Mesa Offshore Trust (MOSH) 8-K Mesa Offshore Trust Announces Trust Income for June 2004

Exhibit 99.1

Mesa-Offshore-Trust

Mesa Offshore Trust Announces Trust Income for June 2004

Mesa Offshore Trust

JPMorgan Chase Bank, Trustee	News
	Release

For Immediate Release

Austin, Texas June 18, 2004 - Mesa Offshore Trust  (OTC symbol-MOSH) announced that there will be no Trust income distribution for the month of June 2004 for Unitholders of record on June 30, 2004.

No royalty income was reported this month, and trust expenses were $34,199.

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, as Trustee
	Mike Ulrich
	1(800) 852-1422
	(512) 479-2562

www.businesswire.com/cnn/mosh.htm

P.O. Box 550 Austin, TX 78789