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

Exhibit 99.1

Mesa-Offshore-Trust

Mesa Offshore Trust Announces Trust Income for January 2005
Mesa Offshore Trust
JPMorgan Chase Bank, N.A., Trustee		News
	Release

For Immediate Release

Austin, Texas January 21, 2005 - Mesa Offshore Trust  (OTC symbol-MOSH) announced that there will be no Trust income distribution for the month of January 2005 for Unitholders of record on January 31, 2005.

No royalty income was reported this month, and trust expenses were $65,634.

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

P.O. Box 550   Austin, TX 78789