SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                   FORM 10-Q

[X]  QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(D) OF THE
     SECURITIES EXCHANGE ACT OF 1934 FOR THE QUARTERLY PERIOD ENDED JUNE 30,
     1998

[ ]  TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(D) OF THE
     SECURITIES EXCHANGE ACT OF 1934 FOR THE TRANSITION PERIOD
     FROM _____________ TO _____________.

                         COMMISSION FILE NUMBER 1-8432

                              MESA OFFSHORE TRUST
             (EXACT NAME OF REGISTRANT AS SPECIFIED IN ITS CHARTER)

         TEXAS                                                   76-6004065
(STATE OF INCORPORATION                                       (I.R.S. EMPLOYER
   OR ORGANIZATION)                                          IDENTIFICATION NO.)

         CHASE BANK OF TEXAS,
        NATIONAL ASSOCIATION
     CORPORATE TRUST DIVISION
           712 MAIN STREET
           HOUSTON, TEXAS                                         77002
(ADDRESS OF PRINCIPAL EXECUTIVE OFFICES)                       (ZIP CODE)

                                 (713) 216-6369
              (REGISTRANT'S TELEPHONE NUMBER, INCLUDING AREA CODE)

     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports) and (2) has been subject to such
filing requirements for the past 90 days. Yes [X]  No [ ]

     Indicate the number of shares outstanding of each of the registrant's
classes of common stock, as of the latest practicable date.

     As of August 10, 1998 -- 71,980,216 Units of Beneficial Interest in Mesa
Offshore Trust.

================================================================================
<PAGE>
                        PART I -- FINANCIAL INFORMATION

ITEM 1.  FINANCIAL STATEMENTS

                              MESA OFFSHORE TRUST

                       STATEMENTS OF DISTRIBUTABLE INCOME
                                  (UNAUDITED)
<TABLE>
<CAPTION>
                                           THREE MONTHS ENDED           SIX MONTHS ENDED
                                                JUNE 30,                    JUNE 30,
                                       --------------------------  --------------------------
                                           1998          1997          1998          1997
                                       ------------  ------------  ------------  ------------
<S>                                    <C>           <C>           <C>           <C>         
Royalty income.......................  $    440,956  $  1,786,585  $  1,135,274  $  2,898,140
Interest income......................        27,824        35,801        55,581        53,413
General and administrative expense...      (144,574)      (30,432)     (206,990)     (650,066)
                                       ------------  ------------  ------------  ------------
     Distributable income............  $    324,206  $  1,791,954  $    983,865  $  2,301,487
                                       ============  ============  ============  ============
     Distributable income per unit...  $      .0045  $      .0248  $      .0137  $      .0319
                                       ============  ============  ============  ============
</TABLE>
               STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS

                                           JUNE 30,        DECEMBER 31,
                                             1998              1997
                                       ----------------  ----------------
                                         (UNAUDITED)

               ASSETS
Cash and short-term investments......  $      2,296,382  $      2,993,764
Interest receivable..................            27,824            32,568
Net overriding royalty interest in
  oil and gas properties.............       380,905,000       380,905,000
Accumulated amortization.............      (380,786,721)     (380,656,800)
                                       ----------------  ----------------
                                       $      2,442,485  $      3,274,532
                                       ================  ================
    LIABILITIES AND TRUST CORPUS
Reserve for Trust expenses...........  $      2,000,000  $      2,000,000
Distributions payable................           324,206         1,026,332
Trust corpus (71,980,216 units of
  beneficial interest
  authorized and outstanding)........           118,279           248,200
                                       ----------------  ----------------
                                       $      2,442,485  $      3,274,532
                                       ================  ================

  (The accompanying notes are an integral part of these financial statements.)

                                       1
<PAGE>
                              MESA OFFSHORE TRUST

                     STATEMENTS OF CHANGES IN TRUST CORPUS
                                  (UNAUDITED)
<TABLE>
<CAPTION>
                                            THREE MONTHS ENDED             SIX MONTHS ENDED
                                                 JUNE 30,                      JUNE 30,
                                       ----------------------------  ----------------------------
                                           1998           1997           1998           1997
                                       ------------  --------------  ------------  --------------
<S>                                    <C>           <C>             <C>           <C>           
Trust corpus, beginning of period....  $    168,742  $    1,006,629  $    248,200  $    1,062,405
Distributable income.................       324,206       1,791,954       983,865       2,301,487
Distributions to unitholders.........      (324,206)     (1,791,954)     (983,865)     (2,301,487)
Amortization of net overriding
  royalty
  interest...........................       (50,463)        (89,627)     (129,921)       (145,403)
                                       ------------  --------------  ------------  --------------
Trust corpus, end of period..........  $    118,279  $      917,002  $    118,279  $      917,002
                                       ============  ==============  ============  ==============
</TABLE>
  (The accompanying notes are an integral part of these financial statements.)

                                       2
<PAGE>
                              MESA OFFSHORE TRUST
                         NOTES TO FINANCIAL STATEMENTS
                                  (UNAUDITED)

NOTE 1 -- TRUST ORGANIZATION

     The Mesa Offshore Trust (the "Trust") was created effective December 1,
1982 when Mesa Petroleum Co., predecessor to Mesa Limited Partnership, which was
the predecessor to MESA Inc., transferred a 99.99% interest in the Mesa Offshore
Royalty Partnership (the "Partnership") to the Trust. The Partnership was
created to receive and hold a 90% net overriding royalty interest (the
"Royalty") in ten producing and nonproducing oil and gas properties located in
federal waters offshore Louisiana and Texas (the "Royalty Properties"). Until
August 7, 1997, MESA Inc. owned and operated its assets through Mesa Operating
Co. ("Mesa"), the operator of the Royalty Properties. Mesa Operating Co. is
also the managing general partner of the Partnership (the Managing General
Partners). On August 7, 1997, MESA Inc. merged with and into Pioneer Natural
Resources Company ("Pioneer") formerly a wholly owned subsidiary of MESA, Inc.
and Parker & Parsley Petroleum Company merged with and into Pioneer Natural
Resources USA, Inc. (successor to Mesa Operating Co.), a wholly owned subsidiary
of Pioneer ("PNR") (collectively, the mergers are referred to herein as the
"Merger"). Subsequent to the Merger, Pioneer owns and operates its assets
through PNR and is also the managing general partner of the Partnership. As used
in this report, the term PNR generally refers to the operator of the Royalty
Properties, unless otherwise indicated.

STATUS OF THE TRUST

     In 1996, PNR drilled five wells from the existing "A" platform on the
South Marsh Island 155 block. PNR recovered all remaining costs related to the
South Marsh Island drilling program as of the February 1997 reporting month. In
addition, during the first quarter of 1997, the Trust recovered approximately
$.5 million in general and administrative expenses paid from the Trust's reserve
fund during the period in which Royalty income was not paid to the Trust,
replenishing the Trust's expense reserve fund balance to $2 million.

     The Trust Indenture provides that the Trust will terminate if the total
amount of cash per year received by the Trust falls below certain levels for
each of three successive years. The December 31, 1997 reserve report prepared
for the Partnership (see the Trust's 1997 Annual Report on Form 10-K) indicates
that 95% of future net revenues will be received by the Trust during the next
four years. As such, it is possible, depending on the timing of future
production, market conditions, the success of future drilling activities, if
any, and other matters, that in 1998 the Trust may commence a period of three
successive years in which annual net royalty income would be below the
termination threshold prescribed in the Indenture, resulting in termination of
the Trust pursuant to the terms discussed above.

NOTE 2 -- BASIS OF PRESENTATION

     The accompanying unaudited financial information has been prepared by Chase
Bank of Texas, National Association (the "Trustee") in accordance with the
instructions to Form 10-Q, and the Trustee believes such information includes
all the disclosures necessary to make the information presented not misleading.
The information furnished reflects all adjustments which are, in the opinion of
the Trustee, necessary for a fair presentation of the results for the interim
periods presented. The financial information should be read in conjunction with
the financial statements and notes thereto included in the Trust's 1997 Annual
Report on Form 10-K.

                                       3
<PAGE>
     The financial statements of the Trust are prepared on the following basis:

          (a)  Royalty income recorded for a month is the Trust's interest in
     the amount computed and paid by PNR to the Partnership for such month
     rather than either the value of a portion of the oil and gas produced by
     PNR for such month or the amount subsequently determined to be 90% of the
     net proceeds for such month;

          (b)  Interest income, interest receivable and distributions payable to
     unitholders include interest to be earned on short-term investments from
     the financial statement date through the next distribution date;

          (c)  Trust general and administrative expenses are recorded in the
     month they accrue;

          (d)  Amortization of the net overriding royalty interest, which is
     calculated on the basis of current royalty income in relation to estimated
     future royalty income, is charged directly to trust corpus since such
     amount does not affect distributable income; and

          (e)  Distributions payable are determined on a monthly basis and are
     payable to unitholders of record as of the last business day of each month.
     However, cash distributions are made quarterly in January, April, July and
     October, and include interest earned from the monthly record dates to the
     date of distribution.

     This basis for reporting Royalty income is considered to be the most
meaningful because distributions to the unitholders for a month are based on net
cash receipts for such month. However, it will differ from the basis used for
financial statements prepared in accordance with generally accepted accounting
principles in several respects. Under such principles, Royalty income for a
month would be based on net proceeds from production for such month without
regard to when calculated or received and interest income would be calculated
only for the periods covered by the financial statements and would exclude
interest from the period end to the date of distribution.

     The instruments conveying the Royalty provide that PNR will calculate and
pay the Partnership each month an amount equal to 90% of the net proceeds for
the preceding month. Generally, net proceeds means the excess of the amounts
received by PNR from sales of oil and gas from the Royalty Properties plus other
cash receipts over operating and capital costs incurred.

                                       4
<PAGE>
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
        OF OPERATIONS

NOTE REGARDING FORWARD-LOOKING STATEMENTS

     This Form 10-Q includes "forward-looking statements" within the meaning
of Section 27A of the Securities Act of 1933, as amended, and Section 21E of the
Securities Exchange Act of 1934, as amended. All statements other than
statements of historical facts included in this Form 10-Q, including without
limitation the statements under "Management's Discussion and Analysis of
Financial Condition and Results of Operations" and Note 1 to the financial
statements of the Trust regarding the future net revenues of the Trust, are
forward-looking statements. Although Pioneer has advised the Trust that it
believes that the expectations reflected in such forward-looking statements are
reasonable, no assurance can be given that such expectations will prove to have
been correct. Important factors that could cause actual results to differ
materially from expectations ("Cautionary Statements") are disclosed in this
Form 10-Q, including without limitation in conjunction with the forward-looking
statements included in this Form 10-Q and in the Trust's Form 10-K. All
subsequent written and oral forward-looking statements attributable to the Trust
or persons acting on its behalf are expressly qualified in their entirety by the
Cautionary Statements.

INFORMATION SYSTEMS FOR THE YEAR 2000

     Pioneer has stated that it will be required to modify its information
systems in order to accurately process data referencing the year 2000. Because
of the importance of occurrence dates in the oil and gas industry, the
consequences of not pursuing these modifications could be very significant to
Pioneer's ability to manage and report operating activities. Pioneer has
contracted with third parties to perform the software programming changes
necessary to correct any existing deficiences. Pioneer has stated that such
programming changes are anticipated to be completed and tested by March 1, 1999.

FINANCIAL REVIEW

     During the second quarter of 1998, the Trust had distributable income of
$324,206, representing $.0045 per unit, as compared to $1,791,954, representing
$.0248 per unit in the second quarter of 1997. The per unit amounts of
distributable income for the second quarter of 1998 and 1997 were earned by
month as follows:

                                         1998       1997
                                       ---------  ---------
April................................  $   .0036  $   .0122
May..................................      .0008      .0080
June.................................      .0001      .0046
                                       ---------  ---------
                                       $   .0045  $   .0248
                                       =========  =========

     Royalty income decreased to $440,956 in the second quarter of 1998 as
compared to $1,786,585 in the second quarter of 1997. The decrease in Royalty
income is primarily due to substantially lower production volumes on South Marsh
Island blocks 155 and 156 and West Delta blocks 61 and 62 and decreased prices
for crude oil, condensate and natural gas liquids in the second quarter of 1998
when compared to the corresponding 1997 period.

                                       5
<PAGE>
     Production volumes for natural gas decreased to 331,591 Mcf in the second
quarter of 1998 from 1,057,041 Mcf in the second quarter of 1997. The average
price received for natural gas was $2.21 per Mcf in the second quarter of 1998
compared to $1.99 per Mcf in the second quarter of 1997.

     Crude oil, condensate and natural gas liquids production decreased to
15,357 barrels in the second quarter of 1998 from 45,707 barrels in the second
quarter of 1997. The average price received for crude oil, condensate and
natural gas liquids was $12.84 per barrel in the second quarter of 1998,
compared to $14.91 per barrel in the second quarter of 1997.

     The decrease in natural gas and crude oil, condensate and natural gas
liquids production for both the six months and the quarter ended June 30, 1998
when compared to the comparable periods of 1997 are primarily attributable to
the cessation of production on the A-6 ST and A-21 wells on South Marsh Island
blocks 155 and 156 during late 1997 and natural production declines on the A-20,
A-22 and A-14 ST wells.

     For the six months ended June 30, 1998, natural gas production volumes
decreased to 805,665 Mcf from 2,692,894 Mcf for the six months ended June 30,
1997. Crude oil, condensate and natural gas liquids production volumes decreased
to 24,440 barrels in the first six months of 1998 as compared to 106,404 barrels
in the first six months of 1997. The decrease in natural gas production and
crude oil, condensate and natural gas liquids production was primarily due to
cessation of production on the 5 S#5 West Delta Block 61/62 and the A-6 ST well
on South Marsh Island block 155.

OPERATIONAL REVIEW

     PNR has advised the Trust that during the second quarter of 1998 its
offshore gas production was marketed under short term contracts at spot market
prices to multiple purchasers, including Columbia Energy Services and Enron Gas
Marketing, and that it expects to continue to market its production under short
term contracts for the foreseeable future. Spot market prices for natural gas in
the second quarter of 1998 were higher than spot market prices in the second
quarter of 1997.

     The amount of cash distributed by the Trust is dependent on, among other
things, the sales prices and quantities of gas, crude oil, condensate and
natural gas liquids produced from the Royalty Properties and the quantities
sold. Substantial uncertainties exist with regard to future gas and oil prices,
which are subject to fluctuations due to the regional supply and demand for
natural gas and oil, production levels and other activities of OPEC and other
oil and gas producers, weather, storage levels, industrial growth, conservation
measures, competition and other variables.

     The Brazos A-39 block experienced a decrease in natural gas production in
the second quarter of 1998 as compared to the second quarter of 1997, primarily
due to natural production decline. The Brazos A-7 block also experienced a
decrease in natural gas production in the second quarter of 1998 compared to the
second quarter of 1997, primarily due to natural production decline. PNR farmed
out a portion of the Brazos A-7 block to another operator and participated at a
10% working interest in the completion of an exploratory gas well that was
drilled in the second quarter of 1997. The No. 5 well encountered gas pay and
was suspended pending completion operations. Production facilities were
installed, and the No. 5 well commenced production late in the second quarter at
a rate of approximately 10 MMcf per day. The combined completion and facility
costs are expected to total $6.6 million ($594,000 net to the Trust).

     The South Marsh Island 155 and 156 blocks experienced a decrease in
production in the second quarter of 1998 as compared to the second quarter 1997,
primarily due to natural production decline, and the cessation of production in
the A6 ST. Production for the block is currently 2.2 MMcf per day and 105
barrels of condensate per day as of August 1998. PNR purchased 3D seismic data
for the South Marsh Island 156 block at a cost of $300,000 ($189,000 net to the
Trust). The data has been evaluated and PNR has no current plans for additional
drilling.

     The West Delta 61 and 62 blocks experienced a decrease in production in the
second quarter of 1998 as compared to the second quarter of 1997, primarily due
to the cessation of production of PNR operated

                                       6
<PAGE>
wells. These wells are currently uneconomic to produce. In portions of West
Delta Block 62, the Trust is receiving royalty income from this property
pursuant to a farmout agreement with another operator. The interest in the
farmout wells which is attributable to the trust consists of a 7.5% overriding
royalty interest. In West Delta Block 61, PNR farmed out portions of the block
to another operator, retaining a 10% (9% net to the Trust) overriding royalty
interest. A new well was drilled in the second quarter which encountered 320 net
feet of pay in 8 Miocene sands below a true vertical depth of 7,500 feet.

     Matagorda Island 624 production decreased in the second quarter of 1998 as
compared to the second quarter of 1997, primarily due to natural production
decline. Gross production for the block is currently 2.6 MMcf per day and 38
barrels of condensate per day as of August 1998.

TERMINATION OF THE TRUST

     The terms of the Mesa Offshore Trust Indenture provide that the Trust will
terminate upon the first to occur of the following events: (1) the total amount
of cash received per year by the Trust for each of three successive years
commencing after December 31, 1987 is less than 10 times one-third of the total
amount payable to the Trustee as compensation for such three year period or (2)
a vote by the unitholders in favor of termination. Because the Trust will
terminate in the event the total amount of cash received per year by the Trust
falls below certain levels, it would be possible for the Trust to terminate even
though some of the Royalty Properties continued to have remaining productive
lives. For information regarding the estimated remaining life of each of the
Royalty Properties and the estimated future net revenues of the Trust based on
information provided by PNR, see the Trust's 1997 Annual Report on Form 10-K.
Upon termination of the Trust, the Trustee will sell for cash all the assets
held in the Trust estate and make a final distribution to unitholders of any
funds remaining after all Trust liabilities have been satisfied. The discussion
set forth above is qualified in its entirety by reference to the Trust Indenture
itself, which is available upon request from the Trustee. Amounts paid to the
Trustee as compensation were $173,000, $123,000 and $149,000 for the years
1997, 1996, and 1995, respectively.

     The December 31, 1997, reserve report prepared for the Partnership (see the
Trust's 1997 Annual Report on Form 10-K) indicates that 95% of future net
revenues will be received by the Trust during the next four years. As such, it
is possible, depending on the timing of future production, market conditions,
the success of future drilling activities, if any , and other matters, that in
1998 the Trust may commence a period of three successive years in which annual
net royalty income would be below the termination threshold prescribed in the
Indenture, resulting in termination of the Trust pursuant to the terms discussed
above.

     The terms of the First Amended and Restated Articles of General Partnership
of the Partnership provide that the Partnership shall dissolve upon the
occurrence of any of the following: (a) December 31, 2030; (b) the election of
the Trustee to dissolve the Partnership; (c) the termination of the Trust; (d)
the bankruptcy of the Managing General Partner; or (e) the dissolution of the
Managing General Partner or its election to dissolve the Partnership; provided
that the Managing General Partner shall not elect to dissolve the Partnership so
long as the Trustee remains the only other partner of the Partnership. In the
event of a dissolution of the Partnership and a subsequent winding up and
termination thereof, the assets of the Partnership (i.e., the Royalty interest)
could either (i) be distributed in kind ratably to the Managing General Partner
and the Trustee or (ii) be sold and the proceeds thereof distributed ratably to
the Managing General Partner and the Trustee. In the event of a sale of the
Royalty and a distribution of the cash proceeds to the Trustee, the Trustee
would make a final distribution to unitholders of such cash proceeds plus any
other cash held by the Trust after the payment of or provision for all
liabilities of the Trust, and the Trust would be terminated.

                                       7
<PAGE>
     The following tables provide a summary of the calculations of the net
proceeds attributable to the Partnership's royalty interest:
<TABLE>
<CAPTION>
                                                     SOUTH
                                        BRAZOS       MARSH        WEST
                                        A-7 AND   ISLAND 155    DELTA 61    MATAGORDA
                                         A-39       AND 156      AND 62     ISLAND 624      TOTAL
                                       ---------  -----------  ----------   ----------   -----------
<S>                                    <C>        <C>          <C>           <C>         <C>        
THREE MONTHS ENDED JUNE 30, 1998:
    Ninety percent of gross
      proceeds.......................  $ 255,981  $   399,775  $  105,088    $168,446    $   929,290
    Less ninety percent of --
      Operating expenditures.........    (67,478)    (193,294)   (178,471)    (28,245)      (467,488)
      Capital costs recovered........     --          --           --            (802)          (802)
      Accrual for future abandonment
         costs and interest on cost
         carryforward................    (11,727)      (1,500)     (5,848)       (925)       (20,000)
                                       ---------  -----------  ----------   ----------   -----------
    Net proceeds (excess costs)......  $ 176,776  $   204,981  $  (79,231)   $138,474    $   441,000
                                       =========  ===========  ==========   ==========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                    $   440,956
                                                                                         ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................        153       13,758      --           1,446         15,357
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Bbl....  $   12.11  $     12.86  $   --        $  12.77    $     12.84
                                       =========  ===========  ==========   ==========   ===========
      Natural gas (Mcf)..............    117,251       98,517      43,814      72,009        331,591
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Mcf....  $    2.17  $      2.26  $     2.40    $   2.08    $      2.21
                                       =========  ===========  ==========   ==========   ===========
    Producing wells..................          3            3           3           1             10
THREE MONTHS ENDED JUNE 30, 1997:
    Ninety percent of gross
      proceeds.......................  $ 357,122  $ 1,817,894  $  296,225    $309,604    $ 2,780,845
    Less ninety percent of --
      Operating expenditures.........    (91,808)    (407,236)   (216,251)    (63,503)      (778,798)
      Capital costs recovered........     --         (173,808)     --          (4,564)      (178,372)
      Accrual for future abandonment
         costs.......................    (25,074)      (1,599)     (8,645)     (1,593)       (36,911)
                                       ---------  -----------  ----------   ----------   -----------
    Net proceeds.....................  $ 240,240  $ 1,235,251  $   71,329    $239,944    $ 1,786,764
                                       =========  ===========  ==========   ==========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                    $ 1,786,585
                                                                                         ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................        274       40,527       1,051       3,855         45,707
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Bbl....  $   17.85  $     14.35  $    17.24    $  19.98    $     14.91
                                       =========  ===========  ==========   ==========   ===========
      Natural gas (Mcf)..............    177,383      627,343     142,638     109,677      1,057,041
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Mcf....  $    1.99  $      1.97  $     1.95    $   2.12    $      1.99
                                       =========  ===========  ==========   ==========   ===========
    Producing wells..................          3            5           4           2             14
</TABLE>
------------

o   The amounts shown are for Mesa Offshore Royalty Partnership.

o   The amounts for the three months ended June 30, 1998 and 1997 represent
    actual production for the periods February 1998 through April 1998 and
    February 1997 through April 1997, respectively.

o   Capital costs recovered represents capital costs incurred during the current
    or prior periods to the extent that such costs have been recovered by PNR
    from current period Gross Proceeds.

o   Producing wells indicate the number of wells capable of production as of the
    end of the period.

                                       8
<PAGE>
<TABLE>
<CAPTION>
                                                       SOUTH
                                         BRAZOS        MARSH         WEST      MATAGORDA
                                         A-7 AND     ISLAND 155    DELTA 61     ISLAND
                                          A-39        AND 156       AND 62        624         TOTAL
                                        ---------    ----------   ----------   ---------   -----------
<S>                                     <C>          <C>          <C>          <C>         <C>        
SIX MONTHS ENDED JUNE 30, 1998:
    Ninety percent of gross
      proceeds.......................   $ 557,803    $  725,015   $  455,209   $ 562,188   $ 2,300,215
    Less ninety percent of --
      Operating expenditures.........    (157,378)     (507,546)    (356,010)   (102,011)   (1,122,945)
      Capital costs recovered........      --            --           --          (1,883)       (1,883)
      Accrual for future abandonment
         costs.......................     (23,454)       (3,000)     (11,696)     (1,850)      (40,000)
                                        ---------    ----------   ----------   ---------   -----------
    Net proceeds (excess costs)......   $ 376,971    $  214,469   $   87,503   $ 456,444   $ 1,135,387
                                        =========    ==========   ==========   =========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                      $ 1,135,274
                                                                                           ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................         433        20,252          594       3,161        24,440
                                        =========    ==========   ==========   =========   ===========
      Average sales price per Bbl....   $   13.71    $    14.17   $    15.51   $   14.80   $     14.28
                                        =========    ==========   ==========   =========   ===========
      Natural gas (Mcf)..............     233,776       182,941      172,719     216,229       805,665
                                        =========    ==========   ==========   =========   ===========
      Average sales price per Mcf....   $    2.36    $     2.39   $     2.58   $    2.38   $      2.42
                                        =========    ==========   ==========   =========   ===========
    Producing wells..................           3             3            3           1            10
<CAPTION>
                                                       SOUTH
                                         BRAZOS        MARSH         WEST       MATAGORDA
                                         A-7 AND     ISLAND 155    DELTA 61      ISLAND
                                          A-39        AND 156       AND 62         624         TOTAL
                                        ---------    ----------   -----------   ---------   ------------
SIX MONTHS ENDED JUNE 30, 1997:
    Ninety percent of gross
      proceeds.......................   $1,042,580   $6,721,943   $   982,972   $ 782,018   $  9,529,513
    Less ninety percent of --
      Operating expenditures.........    (208,538)     (731,787)     (411,572)   (137,629)    (1,489,526)
      Capital costs recovered........      --        (4,948,156)      (33,897)     (4,564)    (4,986,617)
      Accrual for future abandonment
      costs..........................     (85,116)      (18,738)      (42,513)     (8,573)      (154,940)
                                        ---------    ----------   -----------   ---------   ------------
    Net proceeds.....................   $ 748,926    $1,023,262   $   494,990   $ 631,252   $  2,898,430
                                        =========    ==========   ===========   =========   ============
    Trust share of net proceeds
      (99.99%).......................                                                       $  2,898,140
                                                                                            ============
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................         551        98,090         2,504       5,259        106,404
                                        =========    ==========   ===========   =========   ============
      Average sales price per Bbl....   $   19.88    $    17.92   $     19.02   $   20.82   $      18.10
                                        =========    ==========   ===========   =========   ============
      Natural gas (Mcf)..............     383,203     1,735,686       338,177     235,828      2,692,894
                                        =========    ==========   ===========   =========   ============
      Average sales price per Mcf....   $    2.69    $     2.86   $      2.77   $    2.85   $       2.82
                                        =========    ==========   ===========   =========   ============
    Producing wells..................           3             5             4           2             14
</TABLE>
------------

o   The amounts shown are for Mesa Offshore Royalty Partnership.

o   The amounts for the six months ended June 30, 1998 and 1997 represent actual
    production for the periods November 1997 through April 1998, and November
    1996 through April 1997 respectively.

o   Capital costs recovered represents capital costs incurred during the current
    or prior periods to the extent that such costs have been recovered by PNR
    from current period Gross Proceeds.

o   Producing wells indicate the number of wells capable of production as of the
    end of the period.

                                       9
<PAGE>
                                    PART II

ITEM 6.  EXHIBITS AND REPORTS ON FORM 8-K

     (A)  EXHIBITS

     (Asterisk indicates exhibit previously filed with the Securities and
Exchange Commission and incorporated herein by reference.)
<TABLE>
<CAPTION>
                                                                                                 SEC FILE
                                                                                                    OR
                                                                                               REGISTRATION    EXHIBIT
                                                                                                  NUMBER       NUMBER
                                                                                               ------------    -------
<C>                <S>                                                                         <C>             <C>
       4(a)        *Mesa Offshore Trust Indenture between Mesa Petroleum Co. and Texas
                    Commerce Bank National Association, as Trustee, dated December 15,
                    1982....................................................................      2-79673         10(gg)
       4(b)        *Overriding Royalty Conveyance between Mesa Petroleum Co. and Mesa
                    Offshore Royalty Partnership, dated December 15, 1982...................      2-79673         10(hh)
       4(c)        *Partnership Agreement between Mesa Offshore Management Co. and Texas
                    Commerce Bank National Association, as Trustee, dated December 15,
                    1982....................................................................      2-79673         10(ii)
       4(d)        *Amendment to Partnership Agreement between Mesa Offshore Management Co.,
                    Texas Commerce Bank National Association, as Trustee, and Mesa Operating
                    Limited Partnership, dated December 27, 1985 (Exhibit 4(d) to Form 10-K
                    for year ended December 31, 1992 of Mesa Offshore Trust)................       1-8432          4(d)
       4(e)        *Amendment to Partnership Agreement between Texas Commerce Bank National
                    Association, as Trustee and Mesa Operating dated as of January 5, 1994
                    (Exhibit 4(e) to Form 10-K for year ended December 31, 1993 of Mesa
                    Offshore Trust).........................................................       1-8432          4(e)
         27        Financial Data Schedule
</TABLE>
     (B)  REPORTS ON FORM 8-K

          None.

                                       10
<PAGE>
                                   SIGNATURES

     PURSUANT TO THE REQUIREMENTS OF THE SECURITIES EXCHANGE ACT OF 1934, THE
REGISTRANT HAS DULY CAUSED THIS REPORT TO BE SIGNED ON ITS BEHALF BY THE
UNDERSIGNED, THEREUNTO DULY AUTHORIZED.

                                          MESA OFFSHORE TRUST

                                             CHASE BANK OF TEXAS,
                                          By NATIONAL ASSOCIATION
                                             --------------------
                                                  TRUSTEE

                                          By /s/ PETE FOSTER
                                             --------------------
                                                 PETE FOSTER
                                        SENIOR VICE PRESIDENT & TRUST
                                                 OFFICER

Date:  August 10, 1998

     The Registrant, Mesa Offshore Trust, has no principal executive officer,
principal financial officer, board of directors or persons performing similar
functions. Accordingly, no additional signatures are available and none have
been provided.

                                       11