Mesa Offshore Trust (MOSH) 8-K Mesa Offshore Trust Announces Trust Income for June 2005

Exhibit 99.1

Mesa-Offshore-Trust

Mesa Offshore Trust Announces Trust Income for June 2005

Mesa Offshore Trust

JPMorgan Chase Bank, N.A., Trustee	News
	Release

For Immediate Release

Austin, Texas June 20, 2005 - Mesa Offshore Trust (OTCBB symbol-MOSH)

announced that there will be no Trust income distribution for the month of June 2005 for Unitholders of record on June 30, 2005.

Royalty income will total approximately $178,000, and Trust expenditures will be approximately $70,000. The Trust will withhold the difference to recoup expenses incurred by the Trust during prior periods in which there was no royalty income and to increase the reserve for Trust expenses. As of May 31, 2005, the reserve for Trust expenses, excluding interest receivable, was approximately $533,000.

The extent of future distributions from the properties in which the Trust has an interest will continue to be dependent on normal factors associated with oil and gas operations such as oil and gas production levels, prices and associated cost, accruals for future abandonment costs, timing and extent of capital expenditures.

Contact:Mesa Offshore Trust

JPMorgan Chase Bank, N.A., as Trustee

Mike Ulrich

1(800) 852-1422

(512) 479-2562

www.businesswire.com/cnn/mosh.htm

700 Lavaca            Austin, TX 78701