Mesa Offshore Trust (MOSH) 10-Q 2005 Q1 Quarterly report

EXHIBIT 32

May 20, 2005

Via EDGAR

Securities and Exchange Commission
Judiciary Plaza
450 Fifth Street, N.W.
Washington, D.C. 20549

RE:Certification pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002

Ladies and Gentlemen:

In connection with the Quarterly Report of Mesa Offshore Trust (the “Trust”) on Form 10-Q for the quarterly period ended  March 31, 2005, as filed with the Securities and Exchange Commission on the date hereof (the “Report”), the undersigned, not in its individual capacity but solely as the trustee of the Trust, certifies pursuant to 18 U.S.C. 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that to its knowledge:

(1)The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934, as amended; and

(2)The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Trust.

The above certification is furnished solely pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 (18 U.S.C. 1350) and is not being filed as part of the Form 10-Q or as a separate disclosure document.

JPMORGAN CHASE BANK, N.A.,
Trustee for Mesa Offshore Trust

/s/ MIKE ULRICH

Mike Ulrich

Vice President and Trust Officer