Exhibit 10.1

October 6, 2011

To the Consenting Noteholders:

Ladies and Gentlemen:

This restructuring support and lockup letter agreement (together with the exhibits and schedule hereto, this “Agreement”) sets forth certain terms and conditions pursuant to which Dune Energy, Inc. (the “Company”) will restructure its indebtedness by consummating an out-of-court restructuring or a prearranged or prepackaged plan of reorganization in a voluntary case (the “Chapter 11 Case”) to be filed under chapter 11 of title 11 of the United States Code, 11 U.S.C. §§ 101 et seq. (the “Bankruptcy Code”) to be effectuated with the support (as set forth herein) of the undersigned noteholders (the “Consenting Noteholders”) holding at least 75% of the aggregate principal amount of notes (the “Notes”) outstanding under that certain indenture, dated as of May 15, 2007 (as amended to date, and as it may hereafter be amended from time to time, the “Notes Indenture”) among the Company, as issuer, the guarantors thereto and the Bank of New York, as trustee and collateral agent (the “Notes Trustee”) (each of the foregoing, a “Party,” and, collectively, the “Parties”).

Capitalized terms used but not defined herein shall have the meanings ascribed to such terms in the Restructuring Term Sheet (as defined below).

The parties hereto hereby agree as follows:

The principal terms of the proposed restructuring are set forth on the term sheet annexed hereto as Exhibit A (which term sheet is expressly incorporated by reference herein and made binding on the Parties as a part of this Agreement as if fully set forth herein (as such term sheet may be modified in accordance with Section 14 hereof, the “Restructuring Term Sheet”)). The restructuring of the Notes Claims (as defined below) incorporating the terms and conditions set forth herein and in the Restructuring Term Sheet and containing no other provisions adverse to the Consenting Noteholders except as the Company and the Required Consenting Noteholders (as defined below) may expressly consent to in writing is referred to herein as the “Restructuring”.

The Restructuring shall be implemented on a consensual basis, either out-of-court or through the Chapter 11 Case, and will provide for, among other things, treatment of the claims under the Notes Indenture and all related claims, rights and causes of action arising out of or in connection with the Notes and/or Notes Indenture (collectively, the “Note Claims”) as specified in the Restructuring Term Sheet.

Each of the Consenting Noteholders severally and not jointly represents and warrants that, as of the date such Consenting Noteholder executes and delivers this Agreement:

(a) it is the beneficial owner of the face amount of the Notes, or is the nominee, investment manager, or advisor for beneficial holders of the Notes, as reflected in such Consenting Noteholder’s signature block to this Agreement, which amount the Company and each Consenting Noteholder understands and acknowledges is proprietary and confidential to such Consenting Noteholder;

(b) has full power and authority to vote on and consent to such matters concerning the Note Claims and to exchange, assign and transfer such Note Claims;

(c) other than pursuant to this Agreement, such Notes are free and clear of any pledge, lien, security interest, charge, claim, equity, option, proxy, voting restriction, right of first refusal or other limitation on disposition, or encumbrances of any kind, that would adversely affect in any way such Consenting Noteholder’s performance of its obligations contained in this Agreement at the time such obligations are required to be performed; and

(d) (i) it is either (a) a qualified institutional buyer as defined in Rule 144A of the Securities Act of 1933, as amended (the “Securities Act”) or (b) an institutional accredited investor (as defined in Rule 501(a)(1), (2), (3), or (7) under the Securities Act (the “Rules”)) and (ii) any securities acquired by the Consenting Noteholder in connection with the transactions described herein will not have been acquired with a view to distribution.

The Company shall (a) support and use reasonable commercial efforts to complete the Restructuring on the terms set forth in the Restructuring Term Sheet, (b) use reasonable commercial efforts to do all things necessary and appropriate in furtherance of the Restructuring embodied in the Restructuring Term Sheet, including, without limitation, if the Chapter 11 Case is commenced (i) taking all steps reasonably necessary to obtain an order of the United States Bankruptcy Court in which the Chapter 11 Case is filed (the “Bankruptcy Court”), acceptable in all material respects to the Required Consenting Noteholders, confirming a plan of reorganization consistent with the terms set forth in the Restructuring Term Sheet (the “Plan”) within the timeframes contemplated by this Agreement, and (ii) taking all steps reasonably necessary and desirable to cause the effective date of the Plan to occur within the timeframes contemplated by this Agreement, (c) obtain any and all required regulatory and/or third-party approvals for the

2

Restructuring embodied in the Restructuring Term Sheet and (d) except as provided in Section 29, not take any action that is inconsistent with, or is intended or is likely to interfere with consummation of, the Restructuring and the transactions embodied in the Restructuring Term Sheet, including, without limitation, from and after the date hereof, not directly or indirectly, seek, pursue, propose or support, or encourage the pursuit, proposal or support, of any offer to restructure the Company, including proposing a plan of reorganization that is inconsistent with, and could reasonably be expected to prevent, delay or impede, the Restructuring of the Company in accordance with the terms set forth in the Plan.

(a) Subject to the terms and conditions hereof and for so long as no Support Termination Event (as hereinafter defined) shall have occurred, each Consenting Noteholder shall, and shall cause each of its affiliates, subsidiaries, managed funds, representatives, agents and employees, in each case to the extent controlled by such Consenting Noteholder, as applicable, to (i) negotiate in good faith the definitive documentation contemplated by this Agreement or otherwise necessary to effectuate the Restructuring, which shall be in form and substance acceptable to the Required Consenting Noteholders (as defined below), on the terms and subject to the conditions as substantially set forth in this Agreement, (ii) consent to those actions contemplated by this Agreement or otherwise required to be taken to effectuate the Restructuring, including entering into all such documents and agreements, and directing the Notes Trustee to enter into all such documents and agreements, necessary to consummate the Restructuring, (iii) not take any action or otherwise pursue any right or remedy against the Company under applicable law, the Note Claims or the Notes Indenture, as applicable, or initiate, or have initiated on its behalf, any litigation or proceeding of any kind with respect to the Note Claims, including, but not limited to, any action in connection with the December interest payment due under the Notes Indenture, other than to enforce this Agreement (iv) vote its Note Claims to accept both the Out-of-Court Restructuring and the Prepackaged Reorganization (as such terms are defined in the Restructuring Term Sheet) in accordance with the instructions provided in any ballot, which shall be distributed as part of the solicitation of acceptances or rejections of the Restructuring, and shall not change or withdraw (or cause to be changed or withdrawn) such votes; (v) not seek, solicit, support, vote its Note Claims for, or consent to, an Alternative Proposal (as defined below), and (vi) not take any action inconsistent with the transactions expressly contemplated by this Agreement or that would materially delay or obstruct the consummation of the Restructuring, including, without limitation, commencing a voluntary, or joining with any person in commencing, any involuntary, bankruptcy filing by or against the Company. The term “Alternative Proposal” shall mean any plan of reorganization, proposal, offer, dissolution, winding up, liquidation, reorganization, merger, consolidation, business combination, joint venture, partnership, sale of substantially all assets or equity interests, or restructuring (other than the Restructuring) for the Company.

3

(b) If the Company commences the Chapter 11 Case to effectuate the Restructuring, subject to the terms and conditions hereof and for so long as no Support Termination Event (as hereinafter defined) shall have occurred, each Consenting Noteholder agrees (i) that it will timely vote its Note Claims and any other claims or interests that it holds (if it has not done so already) in favor of the Plan and will not revoke or withdraw such vote and (ii) not to (A) object to, delay, impede, or take any other action to interfere with acceptance or implementation of the Plan, including encouraging a third party to take any such actions, or (B) propose, file, support, or vote for any restructuring, workout, or plan of reorganization for the Company other than the Plan, including encouraging a third party to take any such actions, and shall direct the Notes Trustee not to take any action contemplated in (A) and (B) of this Section.

(c) The acceptance of any Consenting Noteholders will not be solicited until such Consenting Noteholder shall have received the applicable documents requesting a vote on the Restructuring.

Each Consenting Noteholder agrees that so long as a Support Termination Event has not occurred, it shall not directly or indirectly (a) grant any proxies to any person in connection with its Note Claims to vote on the Restructuring, or (b) sell, pledge, hypothecate, or otherwise transfer or dispose of, or grant, issue or sell any option, right to acquire, voting participation or other interest in (each, a “Transfer”) any Note Claims, except, in each case, (i) to a transferee that is a Consenting Noteholder, and (ii) to a party that first agrees in writing by executing a joinder in the form of Exhibit B to be subject to the terms and conditions of this Agreement as a “Consenting Noteholder.” Each Consenting Noteholder agrees to notify the Company and counsel to the informal committee of holders of Notes (the “Informal Noteholders Committee”) in writing within five business days after the closing of such trade. Any Transfer of any Note Claims that does not comply with the foregoing shall be deemed void ab initio. This Agreement shall in no way be construed to preclude any Consenting Noteholder or an affiliate of a Consenting Noteholder from acquiring additional Note Claims; provided, however, that any such additional Note Claims acquired by a Consenting Noteholder or an affiliate of a Consenting Noteholder shall, upon acquisition, automatically be deemed to be subject to all the terms of this Agreement.

Each of the Parties represents, warrants, and covenants to the other Parties, as of the date of this Agreement, as follows (each of which is a continuing representation, warranty, and covenant):

(a) Enforceability. It is validly existing and in good standing under the laws of the jurisdiction of its organization, and this Agreement is a legal, valid, and binding obligation of such Party, enforceable against it in accordance with its terms, except as enforcement may be limited by applicable laws relating to or limiting creditor’s rights generally or by equitable principles relating to enforceability.

4

(b) No Consent or Approval. Except as expressly provided in this Agreement or the Bankruptcy Code, no consent or approval is required by any other person or entity in order for it to carry out the Restructuring contemplated by, and perform the respective obligations under, this Agreement.

(c) Power and Authority. Except as expressly provided in this Agreement, it has all requisite power and authority to enter into this Agreement and to carry out the Restructuring contemplated herein, and perform its respective obligations under, this Agreement.

(d) Authorization. The execution and delivery of this Agreement and the performance of its obligations hereunder have been duly authorized by all necessary action on its part.

(e) Representation by Counsel. It has been represented by counsel in connection with this Agreement and the transactions contemplated by this Agreement.

(f) Actions under this Agreement. It is not aware of any event that, due to any fiduciary or similar duty to any other person, would prevent it from taking any action required of it under this Agreement.

If the Company and the Required Consenting Noteholders agree to effectuate the Restructuring through the Chapter 11 Case, the Company shall use its best reasonable efforts to provide draft copies of all “first day” motions or applications and other documents the Company intends to file with the Bankruptcy Court to counsel for the Informal Noteholders Committee at least three business days prior to the date when the Company intends to file such document and shall consult in good faith with such counsel regarding the form and substance of any such proposed filing with the Bankruptcy Court. The Company will use its reasonable best efforts to provide draft copies of all other pleadings the Company intends to file with the Bankruptcy Court to counsel to the Informal Noteholders Committee at least three business days prior to filing such pleading and shall consult in good faith with such counsel regarding the form and substance of any such proposed pleading; provided, however, the Company will not be in breach of this provision by failing to provide to the Informal Noteholders Committee drafts of motions or pleadings that seek emergency or expedited relief.

5

This Agreement shall only become effective (the date on which this Agreement becomes effective, the “Effective Date”) upon satisfaction of the following condition:

(a) (i) the Company has executed and delivered counterpart signature pages of this Agreement to counsel to the Informal Noteholders Committee and (ii) holders of at least 75% in amount of outstanding Note Claims shall have executed and delivered to the Company counterpart signature pages of this Agreement.

(a) This Agreement shall terminate and, except as otherwise provided herein, all obligations of the parties hereto shall immediately terminate and be of no further force and effect upon the occurrence of any of the following (each, a “Support Termination Event”):

(i) the termination of this Agreement by the mutual written consent of the Company and the Consenting Noteholders holding no less than a majority in principal amount of the Notes Claims held at any such time of the Consenting Noteholders (the “Required Consenting Noteholders”);

(ii) the commencement of an involuntary case against the Company under the Bankruptcy Code if such involuntary case is not dismissed within 60 days of it having been commenced (so long as no order for relief is theretofore entered), unless such involuntary case has been converted to a chapter 11 case with the consent of the Company and no other Support Termination Event has occurred;

(iii) the conversion of the Chapter 11 Case to a case under chapter 7 of the Bankruptcy Code;

(iv) the appointment of a trustee, receiver or examiner with expanded powers in the Chapter 11 Case;

(v) the delivery to the Company by the Required Consenting Noteholders of written notice of termination following the occurrence of any of the events listed below:

(A) the Company shall have (1) publicly announced its intention not to pursue the Restructuring, or (2) proposed or accepted an Alternative Proposal;

6

(B) the principal holder of the Preferred Stock fails to enter into the Preferred Stockholder Support Agreement or such agreement is breached or terminated;

(C) the Company fails to launch the solicitation of votes for the Restructuring by November 1, 2011;

(D) the Company fails to consummate the Restructuring on an out-of-court basis by December 15, 2011;

(E) if the Company and the Required Consenting Noteholders agree to effectuate the Restructuring through the Chapter 11 Case:

(I) the Company fails to commence the Chapter 11 Case (the “Petition Date”) on or before (i) November 1, 2011 if the Company does not launch the solicitation of votes or (ii) December 30, 2011 if the Company launches the solicitation of votes, but does not consummate the Restructuring on an out-of-court basis;

(II) the Company fails to file the Plan and related disclosure statement (the “Solicitation Materials”) with the Bankruptcy Court within 5 days after the Petition Date, each of which shall be materially consistent with this Agreement and the Restructuring Term Sheet and shall be in a form and substance acceptable to counsel to the Informal Noteholders Committee;

(III) the Bankruptcy Court’s order(s) approving the Solicitation Materials and setting a hearing to confirm the Plan shall not have been entered by the Bankruptcy Court within 35 days after the filing of the Plan, or as soon thereafter as the Bankruptcy Court’s schedule permits;

(IV) the Bankruptcy Court’s order confirming the Plan (the “Confirmation Order”), which Plan, including all exhibits, appendices, plan supplement documents, and related documents, shall be acceptable to counsel to the Informal Noteholders Committee, shall not have been entered by the Bankruptcy Court within 35 days after the date that the Solicitation Materials are approved; provided, however, that so long as the Company is proceeding in good faith towards confirmation of the Plan, upon written notice from the Company to counsel to the

7

Informal Noteholders Committee in accordance with Section 24 hereof, there shall be a 15-day extension of such 35-day period;

(V) the effective date of the Plan shall not have occurred within 15 days after the date that the Plan is confirmed; provided, however, that so long as the Company is proceeding in good faith towards confirmation of the Plan, upon written notice from the Company to counsel to the Informal Noteholders Committee in accordance with Section 24 hereof, there shall be a 15-day extension of such 15-day period;

(VI) the amendment, modification, or filing of a pleading by the Company seeking to amend or modify the Plan, Solicitation Materials, or any documents related to the foregoing, including motions, notices, exhibits, appendices, and orders, in a manner not reasonably acceptable to counsel to the Informal Noteholders Committee;

(F) the Company or any of their advisors fail to reasonably cooperate with the Informal Noteholders Committee or their professionals or provide them upon request at any time or from time to time with reasonable access to information regarding the operations, business affairs and financial condition of the Company, as requested by the Informal Noteholders Committee and their professionals; provided, however, any written notice of termination based on this provision shall specify what actions the Company or its advisors would need to take to reasonably cooperate and that taking such actions would be deemed to cure this Support Termination Event;

(G) the breach in any material respect by the Company of any of the obligations, representations, warranties, or covenants of the Company set forth in this Agreement;

(H) the issuance by any governmental authority, including any regulatory authority or court of competent jurisdiction, of any ruling or order enjoining the consummation of the Restructuring in a way that cannot be reasonably remedied by the Company in a manner that does not prevent or diminish in a material way compliance with the terms of the Restructuring Term Sheet and this Agreement; provided, however, that the Company shall

8

have ten business days after receiving such ruling or order to cure any breach in a manner that does not prevent or diminish in a material way compliance with the terms of the Restructuring Term Sheet and this Agreement; or

(vi) the delivery of written notice of termination from the Company to counsel for the Informal Noteholders Committee upon the occurrence of any of the following: (A) any material breach, in the aggregate, of one or more Consenting Noteholders’ representations, warranties, covenants, or agreements set forth in this Agreement if such breach is continuing and would have a material adverse effect on the Company’s ability to complete the Restructuring; (B) the board of directors of the Company reasonably determines based upon the advice of counsel that proceeding with the Restructuring would be inconsistent with the exercise of its fiduciary duties; (C) the issuance by any governmental authority, including any regulatory authority or court of competent jurisdiction, of any ruling or order enjoining the consummation of a material portion of the Restructuring. The Company, its directors and its officers shall have no liability to any of the Consenting Noteholders in respect of any termination of this Agreement in accordance with the terms of this Section 9(a)(v); or (D) the principal holder of the Preferred Stock fails to enter into the Preferred Stockholder Support Agreement or such agreement is breached or terminated.

(b) No Support Termination Event shall occur if the Company’s failure to accomplish the actions set forth above is due to or caused by a breach or violation by Consenting Noteholders of their covenants and agreements set forth herein. The Company shall have five business days from delivery of written notice of termination from the Required Consenting Noteholders to the Company to cure any Support Termination Event set forth in this Section 9(a)(v)(A) – (H).

(c) Upon the occurrence of a Support Termination Event, this Agreement shall be of no further force and effect and each Party hereto shall be released from its commitments, undertakings, and agreements under or related to this Agreement and shall have the rights and remedies that it would have had it not entered into this Agreement, and shall be entitled to take all actions, whether with respect to the Restructuring or otherwise, that it would have been entitled to take had it not entered into this Agreement. Upon the occurrence of any termination of this Agreement, any and all consents tendered by the Consenting Noteholders prior to such termination shall be deemed, for all purposes, to be null and void from the first instance and shall not be considered or otherwise used in any manner by the Parties in connection with the Restructuring and this Agreement or otherwise.

9

(d) Notwithstanding any provision in this Agreement to the contrary, upon the written consent of the Required Consenting Noteholders, the dates set forth in this Section may be extended prior to or upon each such date and such later dates agreed to in lieu thereof and shall be of the same force and effect as the dates provided herein.

To the extent any Consenting Noteholder acquires additional Note Claims or holds or acquires any other claims against the Company entitled to vote on the Restructuring, each such Consenting Noteholder agrees that such Note Claims or other claims shall be subject to this Agreement and that it shall vote (or cause to be voted) any such additional Note Claims or other claims, (to the extent still held by it or on its behalf at the time of such vote) to accept the Restructuring.

This Agreement shall terminate automatically without any further required action or notice on the date that the Restructuring is consummated.

It is understood and agreed by the parties hereto that money damages would not be a sufficient remedy for any breach of this Agreement by any party and each non-breaching party shall be entitled to seek specific performance and injunctive or other equitable relief as a remedy for any such breach, and each party agrees to waive any requirement for the securing or posting of a bond in connection with such remedy, in addition to any other remedy to which such non-breaching party may be entitled, at law or in equity. In no event shall any party hereto be liable for any special, indirect, incidental, punitive or consequential damages of any kind or nature whatsoever.

This Agreement constitutes the full and entire understanding and agreement among the parties hereto with regard to the subject matter hereof. This Agreement supersedes all prior negotiations, and documents reflecting such prior negotiations, between and among the Company and the Consenting Noteholders (and their respective advisors), with respect to the subject matter hereof.

No amendment, modification, waiver or supplement of or to the terms of this Agreement shall be valid unless such amendment, modification, waiver or supplement is in writing and has been signed by the Company and the Consenting Noteholders holding no less than two-thirds in principal amount of the Note Claims held at any such time of the Consenting Noteholders.

10

Each Consenting Noteholder hereby confirms that it (i) has conducted an independent review and analysis of the business and affairs of the Company that it considers sufficient and reasonable for purposes of entering into this Agreement, (ii) has had the opportunity to ask questions of, and receive answers from, the Company, and the Company’s representatives, concerning the terms and conditions of the Restructuring and the Restructuring Term Sheet, (iii) is not relying on the Company, or any of its representatives for advice with respect to the legal, tax, economic and related considerations of the Restructuring and the Restructuring Term Sheet, and each Consenting Noteholder has relied on the advice of, or has consulted with, only its own advisers with respect to such matters, (iv) has based its decision to execute this Agreement on its independent review and analysis and (v) has not relied, and is not relying, on the recommendation or advice of the Notes Trustee (or any of its representatives), or the Company (or any of its representatives) or any other person in connection with such decision.

THIS AGREEMENT IS TO BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH THE LAWS OF THE STATE OF NEW YORK APPLICABLE TO CONTRACTS MADE AND TO BE PERFORMED IN SUCH STATE, WITHOUT GIVING EFFECT TO THE CONFLICT OF LAWS PRINCIPLES THEREOF. Each Party hereto agrees that it shall bring any action or proceeding in respect of any claim arising out of or related to this Agreement, to the extent possible, in either the United States District Court for the Southern District of New York or any New York State court sitting in the Borough of Manhattan (the “Chosen Courts”), and solely in connection with claims arising under this Agreement (a) irrevocably submits to the exclusive jurisdiction of the Chosen Courts, (b) waives any objection to laying venue in any such action or proceeding in the Chosen Courts, and (c) waives any objection that the Chosen Courts are an inconvenient forum or do not have jurisdiction over any Party hereto; provided, however, upon the commencement of the Chapter 11 Case, each of the Parties agree that the Bankruptcy Court presiding over the Chapter 11 Case shall be the sole Chosen Court. Each Party hereto irrevocably waives any and all right to trial by jury in any legal proceeding arising out of or relating to this Agreement or the transactions contemplated hereby.

This Agreement is intended to bind and inure to the benefit of the Parties and their respective permitted successors and assigns. Nothing in this Agreement, express or implied, shall give to any person or entity, other than the Parties, any benefit or any legal or equitable right, remedy, or claim under this Agreement. The invalidity or unenforceability at any time of any provision hereof in any jurisdiction shall not affect or diminish in any way the continuing validity and enforceability of the remaining provisions hereof or the continuing validity and enforceability of such provision in any other jurisdiction and any such prohibited or unenforceable provision shall be deemed

11

reformed and construed so that it will be valid, legal, and enforceable and not prohibited to the maximum extent permitted by applicable law. The agreements, representations and obligations of the Consenting Noteholders under this Agreement are, in all respects, several and not joint.

Unless expressly stated herein, this Agreement shall be solely for the benefit of the Parties, and no other person or entity shall be a third-party beneficiary hereof.

This Agreement may be executed in several counterparts, each of which shall be deemed to be an original, and all of which together shall be deemed to be one and the same agreement. Execution copies of this Agreement may be delivered by facsimile, electronic mail or otherwise, each of which shall be deemed to be an original for the purposes of this paragraph.

The section headings of this Agreement are for convenience of reference only and shall not, for any purpose, be deemed a part of this Agreement.

This Agreement is the product of negotiations between the Company and the Informal Noteholders Committee and in the enforcement or interpretation hereof, is to be interpreted in a neutral manner, and any presumption with regard to interpretation for or against any Party by reason of that Party having drafted or caused to be drafted this Agreement, or any portion hereof, shall not be effective in regard to the interpretation hereof.

Notwithstanding anything herein to the contrary, if any Consenting Noteholder is appointed to and serves on an unsecured official committee of creditors in the Chapter 11 Case, the terms of this Agreement shall not be construed so as to limit such Consenting Noteholder’s exercise (in its sole discretion) of its fiduciary duties to any person arising from its service on such committee, and any such exercise (in the sole discretion of such Consenting Noteholder) of such fiduciary duties shall not be deemed to constitute a breach of the terms of this Agreement; provided, however, that nothing in this Agreement shall be construed as requiring any Consenting Noteholder to serve on any official committee in the Chapter 11 Case.

12

It is understood and agreed that no Consenting Noteholder has any fiduciary duty or other duty of trust or confidence in any form with any other Consenting Noteholder, and, except as provided in this Agreement, there are no commitments among or between them.

Any notices, requests and demands required to be provided under this Agreement shall be in writing and shall be deemed to have been duly given or made when actually received at the following addresses as listed below:

13

The Company will afford the Consenting Noteholders and their respective attorneys, consultants, accountants, and other authorized representatives reasonable access, upon reasonable notice during normal business hours, to all properties, books, contracts, commitments, records, management personnel, lenders, and advisors of the Company; provided, however, that the Company’s obligation hereunder shall be conditioned upon such Consenting Noteholder being party to an executed confidentiality agreement approved by and with the Company.

Except as expressly provided in this Agreement, nothing herein is intended to, or does, in any manner waive, limit, impair, or restrict any right of any Consenting Noteholder or the ability of each Consenting Noteholder to protect and preserve its rights, remedies, and interests, including, without limitation, its claims against or interests in the Company. If the Restructuring is not consummated, or if this Agreement is terminated for any reason (other than Section 11 hereof, in which case the Parties shall have any rights set forth in the confirmed Plan, any other court-approved documents, and any other agreements between the Parties entered into after the Petition Date), the Parties fully reserve any and all of their rights. Pursuant to Rule 408 of the Federal Rules of Evidence and any other applicable rules of evidence, this Agreement and all negotiations relating hereto shall not be admissible into evidence in any proceeding other than a proceeding to enforce its terms.

All rights, powers, and remedies provided under this Agreement or otherwise available in respect hereof at law or in equity shall be cumulative and not alternative, and the exercise of any right, power, or remedy thereof by any Party shall not preclude the simultaneous or later exercise of any other such right, power, or remedy by such Party.

14

The Company will submit to the Consenting Noteholders all press releases and public filings relating to this Agreement or the transactions contemplated hereby and any amendments thereof reasonably prior to their release or filing as the case may be. The Company shall not (a) use the name of any Consenting Noteholder in any press release without such Consenting Noteholder’s prior written consent or (b) disclose the holdings of such Consenting Noteholder to any person; provided, however, that the Company shall be permitted to disclose at any time the aggregate principal amount of and aggregate percentage of Note Claims held by the Consenting Noteholders or as otherwise required by the Bankruptcy Code, the Federal Rules of Bankruptcy Procedure or order of the Bankruptcy Court.

Notwithstanding anything to the contrary herein, nothing in this Agreement shall require the Company or any directors or officers of the Company (in such person’s capacity as a director or officer of the Company) to take any action, or to refrain from taking any action, to the extent required to comply with its or their fiduciary or similar duties or obligations under applicable law. Nothing in this Agreement or any document related thereto imposes any liability for actions taken, or not taken, in order to discharge fiduciary obligations described in this Section 29.

The parties hereto shall cooperate with each other in good faith in respect of matters concerning the implementation and consummation of the Restructuring. Subject to the other terms of this Agreement, the Parties agree to execute and deliver such other instruments and perform such acts, in addition to the matters herein specified, as may be reasonably appropriate or necessary, from time to time, to effectuate the Restructuring in a manner materially consistent with the terms set forth in the Restructuring Term Sheet, as applicable.

15

IN WITNESS WHEREOF, the parties hereto have caused this Agreement to be executed and delivered by their respective duly authorized officers, solely in their respective capacity as officers of the undersigned and not in any other capacity, as of the date first set forth above.

EXHIBIT A

RESTRUCTURING TERM SHEET

DUNE ENERGY, INC.

Summary of Terms

October 6, 2011

This preliminary term sheet (this “Term Sheet”) sets forth a brief summary of the principal terms of a potential restructuring (the “Restructuring”) of the capital structure of Dune Energy, Inc. (the “Company”). This Term Sheet is offered in the nature of a settlement proposal in furtherance of settlement discussions, and is intended to be entitled to the protections of Federal Rule of Evidence 408 and any other applicable statutes or doctrines protecting the use or disclosure of confidential information and information exchanged in the context of settlement discussions. This Term Sheet is for discussion only, is a non-binding expression of intent, is intended as an outline only of certain material terms of the proposed transactions described herein, and does not represent a commitment to lend, invest or provide financing or to negotiate to do any of these things. Furthermore, this Term Sheet does not constitute a waiver by any party, or an agreement or commitment by any party to forbear from taking any remedies to which such party may be entitled, and in that regard all rights of the holders (the “Noteholders”) of the Company’s 10-1/2% Senior Secured Notes due 2012 (the “Notes”) are specifically reserved. This Term Sheet and the terms, conditions and assumptions contained herein are subject to the negotiation and execution of definitive documentation for the transactions described herein, including the Restructuring, which documentation shall be in all respects materially consistent with this Term Sheet. This Term Sheet is not an offer with respect to any securities or a solicitation of acceptances of a chapter 11 plan. Any such offer or solicitation will only be made in compliance with all applicable laws.

20

21

22

23

24

25

EXHIBIT B

JOINDER

JOINDER

This Joinder to the Agreement, dated as of April     , 2011, by the Company and the Consenting Noteholders thereto, (the “Agreement”), is executed and delivered by [            ] (the “Joining Party”) as of [            ], 2011 in connection with the transfer from a Consenting Noteholder party to the Agreement to the Joining Party. Capitalized terms used but not defined herein shall have the meanings ascribed to such terms in the Agreement.

1. Agreement to be Bound. The Joining Party hereby agrees to be bound by all of the terms of the Agreement, which is attached to this Joinder as Annex I (as the same may be hereafter amended, restated, or otherwise modified from time to time) as if the Joining Party were an original signatory to the Agreement. From and after the date hereof, the Joining Party shall hereafter be deemed to be a “Consenting Noteholder” for all purposes under the Agreement.

2. Representations and Warranties. With respect to the amount of Note Claims set forth below its name on the signature page hereof and all related claims, rights, and causes of action arising out of or in connection with or otherwise relating to such Claim, the Joining Party hereby makes the representations and warranties of such Consenting Noteholder set forth in the “Representations of the Consenting Noteholders” section of the Agreement to each other Party to the Agreement.

3. GOVERNING LAW; JURISDICTION; JURY TRIAL WAIVER. THIS AGREEMENT IS TO BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH THE LAWS OF THE STATE OF NEW YORK APPLICABLE TO CONTRACTS MADE AND TO BE PERFORMED IN SUCH STATE, WITHOUT GIVING EFFECT TO THE CONFLICT OF LAWS PRINCIPLES THEREOF. Each Party hereto agrees that it shall bring any action or proceeding in respect of any claim arising out of or related to this Agreement, to the extent possible, in either the United States District Court for the Southern District of New York or any New York State court sitting in the Borough of Manhattan (the “Chosen Courts”), and solely in connection with claims arising under this Agreement (a) irrevocably submits to the exclusive jurisdiction of the Chosen Courts, (b) waives any objection to laying venue in any such action or proceeding in the Chosen Courts, and (c) waives any objection that the Chosen Courts are an inconvenient forum or do not have jurisdiction over any Party hereto; provided, however, upon the commencement of the Chapter 11 Case, each of the Parties agree that the Bankruptcy Court presiding over the Chapter 11 Case shall be the sole Chosen Court. Each Party hereto irrevocably waives any and all right to trial by jury in any legal proceeding arising out of or relating to this Agreement or the transactions contemplated hereby.

IN WITNESS WHEREOF, the Joining Party has caused this Joinder to be executed as of the date first written above.