Exhibit 99.1

TEL OFFSHORE TRUST

THE BANK OF NEW YORK MELLON TRUST COMPANY, N.A., CORPORATE TRUSTEE

919 CONGRESS AVENUE / (800) 852-1422 / AUSTIN, TEXAS 78701

TEL OFFSHORE TRUST ANNOUNCES THERE WILL BE NO SECOND QUARTER 2012 DISTRIBUTION

AUSTIN, TEXAS JUNE 28, 2012—TEL OFFSHORE TRUST announced that there will be no trust distribution for the second quarter of 2012 for unitholders of record on June 29, 2012. The Trust has not been able to make a distribution to unitholders for fourteen consecutive quarters, or since January 9, 2009. The financial and operating information included herein for the Trust’s second quarter of 2012 reflects financial and operating information with respect to the royalty properties for the months of February, March and April 2012.

Gas revenues recorded by Chevron U.S.A. Inc. (“Chevron”), as a Working Interest Owner, on its royalty properties decreased approximately 70% to $49,716 in the second quarter of 2012 from $165,069 in the first quarter of 2012.  Natural gas volumes during the second quarter of 2012 decreased approximately 58% to 20,159 Mcf from 48,016 Mcf during the first quarter of 2012. The average price received for natural gas decreased approximately 28% to $2.47 per Mcf in the second quarter of 2012 as compared to $3.44 per Mcf in the first quarter of 2012.

Crude oil revenues recorded by Chevron, as a Working Interest Owner, on its royalty properties decreased approximately 49% to $2,012,960 in the second quarter of 2012 from $3,948,068 in the first quarter of 2012. Oil volumes during the second quarter of 2012 decreased approximately 50% to 17,710 barrels, compared to 35,698 barrels of oil produced in the first quarter of 2012. The average price received for oil increased approximately 3% to $113.66 per barrel in the second quarter of 2012 as compared to $110.60 per barrel in the first quarter of 2012.

Capital expenditures increased by $1,034,632 in the second quarter of 2012 to $1,341,549, as compared to $306,917 in the first quarter of 2012. The increase in capital expenditures is primarily associated with facility improvement projects at Ship Shoal 182/183. Operating expenses decreased by $4,051,989 in the second quarter of 2012 to $3,160,991 as compared to $7,212,980 for the first quarter of 2012. Chevron has informed the Trust that the estimate of the Trust’s net portion of the aggregate cost to plug and abandon the wells subject to the overriding royalty interest on Eugene Island 339 as of the first quarter 2012 is approximately $18.7 million, approximately $18.5 million of which had been incurred through second quarter 2012.

For the second quarter of 2012, under the terms of the conveyance for the royalty properties, production costs for the royalty properties exceeded gross proceeds thereof, with the portion of such excess attributable to the Trust’s remaining 20% royalty interest, equal to approximately $531,862. As of April 30, 2012, aggregate development and production costs for the royalty properties since November 2008 have exceeded the related proceeds of production from the royalty properties by approximately $5.98 million, net to the Trust’s remaining 20% royalty interest.  As a result, the Trust will not be receiving any net proceeds for the second quarter of 2012.  In the fourth quarter of 2010, Chevron U.S.A. Inc. withdrew

$4,304,894 from the Special Cost Escrow account of the Working Interest Owners (a reserve fund for certain costs) to cover expenses incurred in connection with the plugging and abandonment of Eugene Island 339, which served to reduce the amount by which development and production costs exceeded the related proceeds from production; however, additional deposits to the Special Cost Escrow account will be required in future periods in accordance with the terms of the underlying conveyance of the royalty if, and when, net proceeds would otherwise be payable on the royalty.  In the second quarter of 2012, no dollars were escrowed or released from the Special Cost Escrow account of the Working Interest Owners, leaving a balance of $1,000 in the Special Cost Escrow account.

As previously reported, Chevron, as the Managing General Partner of the TEL Offshore Trust Partnership, consummated the sale by the Partnership of 20% of the Partnership’s overriding royalty interest (a 25% net profits interest in certain oil and gas properties), as the Trust needed funds to pay for liabilities of the Trust. The Partnership retained a 20% royalty interest (representing 80% of the original 25%  interest).

This press release contains forward-looking statements.  Although the Managing General Partner of the TEL Offshore Trust Partnership has advised the Trust that the Managing General Partner believes that the expectations contained in this press release are reasonable, no assurances can be given that such expectations will prove to be correct. The Working Interest Owners alone control historical operating data, and handle receipt and payment of funds relating to the royalty properties and payments to the Trust for the related royalty. The Trustees of the Trust cannot assure that errors or adjustments by the Working Interest Owners, whether historical or future, will not affect future royalty income and distributions by the Trust.  Other important factors that could cause these statements to differ materially include delays and costs in connection with repairs or replacements of hurricane-damaged facilities and pipelines, including third-party transportation systems, the actual results of drilling operations, risks inherent in drilling and production of oil and gas properties, and other factors described in the Trust’s Form 10-K for the year ended December 31, 2011 under “Item 1A. Risk Factors,” as well as other risks identified from time to time in its reports on Form 10-Q and Form 8-K as filed with the Securities and Exchange Commission.  Statements made in this press release are qualified by the cautionary statements made in these risk factors. The Trust does not intend, and assumes no obligations, to update any of the statements included in this press release.

The Bank of New York Mellon Trust Company, N.A.

AS CORPORATE TRUSTEE

CONTACT:  Mike Ulrich

(800) 852-1422

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