Mesa Offshore Trust (MOSH) 8-K Mesa Offshore Trust Announces No Trust Income for May 2006

Exhibit 99.1

Mesa-Offshore-Trust

Mesa Offshore Trust Announces No Trust Income for May 2006

MESA OFFSHORE TRUST

JPMorgan Chase Bank, N.A., Trustee

NEWS

RELEASE

For Immediate Release

Austin, Texas May 19, 2006 - Mesa Offshore Trust (OTC symbol-MOSH)

announced that there will be no Trust income distribution for the month of May 2006 for Unitholders of record on May 31, 2006.

Royalty income will total approximately $63,000 and Trust expenditures will be approximately $90,000. Trust expenditures in excess of royalty income received will reduce the Trust’s reserve for Trust expenses. As of April 28, 2006 the reserve for Trust expenses, excluding interest receivable, was approximately $1,377,000.

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