Exhibit 10.2

October 6, 2011

To UBS Securities:

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 UBS Securities LLC (“UBS”) holding approximately 64% of the 10% Senior Redeemable Convertible Preferred Stock of the Company (the “Preferred Stock”) issued under the Certificate of Designation of 10% Senior Redeemable Convertible Preferred Stock of Dune Energy, Inc., Article Fourth of the Amended and Restated Certificate of Incorporation of Dune Energy, Inc. and the resolution of the board of directors of the Company adopted on May 1, 2007 (the Company and UBS each, 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 Company’s indebtedness and equity, including the Preferred Stock, incorporating the terms and conditions set forth herein and in the Restructuring Term Sheet and containing no other provisions adverse to UBS except as the Company and UBS 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 Preferred Stock as specified in the Restructuring Term Sheet.

UBS represents and warrants that, as of the date UBS delivers this Agreement:

(a) it is the beneficial owner of the Preferred Stock as reflected in UBS’ signature block to this Agreement, which amount the Company and UBS understands and acknowledges is proprietary and confidential to UBS;

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

(c) other than pursuant to this Agreement, the Preferred Stock is 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 UBS’ 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 UBS 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 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”) confirming a plan of reorganization consistent with the terms set forth in the Restructuring Term Sheet (the “Plan”), (c) obtain any and all required regulatory and/or third-party approvals for the Restructuring embodied in the Restructuring Term Sheet and (d) except as provided in Section 24, 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.

(a) Subject to the terms and conditions hereof and for so long as no Support Termination Event (as hereinafter defined) shall have occurred, UBS shall, and shall cause each of its affiliates, subsidiaries, managed funds, representatives, agents and employees, in each case to the extent controlled by UBS, as applicable, to (i) negotiate in good faith the definitive documentation contemplated by this Agreement or otherwise

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necessary to effectuate the Restructuring 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, (iii) not take any action or otherwise pursue any right or remedy against the Company under applicable law or initiate, or have initiated on its behalf, any litigation or proceeding of any kind with respect to the Preferred Stock, other than to enforce this Agreement (iv) vote its Preferred Stock 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 Preferred Stock 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. “Alternative Proposal” means 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.

(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, UBS agrees (i) that it will timely vote its Preferred Stock 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.

(c) The acceptance of UBS will not be solicited until UBS shall have received the applicable documents requesting a vote on the Restructuring.

UBS 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

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Preferred Stock 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”) the Preferred Stock.

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.

(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.

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 UBS and (ii) UBS holds at least 64% in amount of issued Preferred Stock and has executed and delivered counterpart signature pages of this Agreement to the Company.

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(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 UBS;

(ii) the termination of the Restructuring Support Agreement among the Consenting Noteholders and the Company dated on or about October 6, 2011;

(iii) the delivery to the Company by UBS of written notice of termination following the occurrence of any of the events listed below:

(A) 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;

(B) the issuance by any governmental authority, including any regulatory authority or court of competent jurisdiction, of any ruling or order permanently 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 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

(iv) the delivery of written notice of termination from the Company to UBS upon the occurrence of any of the following: (A) any material breach of UBS’ 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; or (C) the issuance by any governmental authority,

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including any regulatory authority or court of competent jurisdiction, of any ruling or order permanently enjoining the consummation of a material portion of the Restructuring. The Company, its directors and its officers shall have no liability to UBS in respect of any termination of this Agreement in accordance with the terms of this Section 8(a)(iv).

(b) 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 UBS 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.

To the extent UBS (a) acquires additional Preferred Stock or (b) holds or acquires any other claims against or equity interests in the Company entitled to vote on the Restructuring, UBS agrees that such other claims or equity interests shall be subject to this Agreement and that it shall vote (or cause to be voted) any such additional other claims and equity interests (in each case, 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.

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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 UBS (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 UBS.

UBS 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 UBS 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 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.

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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 reformed and construed so that it will be valid, legal, and enforceable and not prohibited to the maximum extent permitted by applicable law.

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 UBS 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.

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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:

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 UBS or the ability of UBS 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 10 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

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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.

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 24.

Without limiting the generality or effect of the foregoing, notwithstanding any provision of this Agreement or any document related thereto to the contrary, the Company may furnish information to or conduct any discussions with any other person or entity or, upon notice given hereunder, may terminate this Agreement and any other agreement between the Parties (or any of them) if the Company determine in good faith after consultation with outside counsel that doing so is necessary to comply with their fiduciary duties under applicable law.

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.

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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

(Attached)

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.

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