================================================================================

                       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 MARCH 31, 1999

[ ] 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)

<TABLE>
<S>                                                        <C>
                          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)
</TABLE>

                                 1-800-852-1422
              (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 May 12, 1999 -- 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
                                              MARCH 31,
                                       -----------------------
                                         1999         1998
                                       ---------  ------------
<S>                                    <C>        <C>
Royalty income.......................  $  --      $    694,318
Interest income......................     19,148        27,757
General and administrative expense...    (19,148)      (66,037)
                                       ---------  ------------
     Distributable income............  $  --      $    656,038
                                       =========  ============
     Distributable income per unit...  $  --      $      .0091
                                       =========  ============
</TABLE>

               STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS
<TABLE>
<CAPTION>
                                          MARCH 31,         DECEMBER 31,
                                             1999               1998
                                       ----------------     -------------
<S>                                    <C>                  <C>
                                         (UNAUDITED)

<CAPTION>
               ASSETS
<S>                                    <C>                  <C>
Cash and short-term investments......  $      1,764,874     $   1,836,398
Interest receivable..................            19,148            23,102
Net overriding royalty interest in
  oil and gas properties.............       380,905,000       380,905,000
Accumulated amortization.............      (380,848,599)     (380,848,599)
                                       ----------------     -------------
                                       $      1,840,423     $   1,915,901
                                       ================     =============
<CAPTION>
    LIABILITIES AND TRUST CORPUS
<S>                                    <C>                  <C>
Reserve for Trust expenses...........  $      1,784,022     $   1,859,000
Distributions payable................         --                 --
Trust corpus (71,980,216 units of
  beneficial interest
  authorized and outstanding)........            56,401            56,401
                                       ----------------     -------------
                                       $      1,840,423     $   1,915,901
                                       ================     =============
</TABLE>

  (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
                                               MARCH 31,
                                       --------------------------
                                           1999          1998
                                       ------------  ------------
<S>                                    <C>           <C>
Trust corpus, beginning of period....  $     56,401  $    248,200
Distributable income.................       --            656,038
Distributions to unitholders.........       --           (656,038)
Amortization of net overriding
  royalty interest...................       --            (79,458)
                                       ------------  ------------
Trust corpus, end of period..........  $     56,401  $    168,742
                                       ============  ============
</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 is also the
managing general partner of the Partnership (the "Managing General Partner").
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) 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

     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 "Termination Threshold"). The December 31,
1998 reserve report prepared for the Partnership (see the Trust's 1998 Annual
Report on Form 10-K) indicates that Royalty income expected to be received by
the Trust in 1999 and 2000 could be at or near the Termination Threshold. The
reserve report estimates that future Royalty income to the Trust is
approximately $5.2 million while the Termination Threshold for 1998 was
approximately $1.4 million. It is therefore possible (depending on the timing of
future production and drilling activities, market conditions, recoupment of
unrecovered capital costs, the receipt of amounts withheld from the Trust
related to MMS royalty claims, and other matters) that in either 1999 or 2000
Royalty income received by the Trust may be below the Termination Threshold. If
Royalty income falls below the Termination Threshold for three successive years,
the Trust would terminate. 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. There are numerous uncertainties inherent in estimating and
projecting the quantity and value of proved reserves for the Trust properties as
many of the Trust properties are nearing the end of their productive lives and
are therefore subject to unforeseen changes in production rates. As such, there
can be no assurance that Royalty income received by the Trust in 1999 or 2000
will be above the Termination Threshold.

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 1998 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 statements' date through the next distribution date;

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

          (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
     or such other day as the Trustee determines is required to comply with
     legal or stock exchange requirements. 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, these statements differ from 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 include interest earned during
the period 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.

STATUS OF PENDING COSTILLA SALE AGREEMENT

     On April 15, 1999, Pioneer Natural Resources Company announced the
termination of the transaction contemplated by the April 1, 1999 Purchase and
Sales Agreement between PNR and Costilla Energy, Inc. providing for the sale of
certain oil and gas properties to Costilla. Included with the properties to be
sold were PNR's interest in all of the Royalty properties. PNR has executed a
contingency plan to remarket these properties. If such a sale is consummated,
the Trust has been advised that there should be no significant impact on the
Trust, although the precise nature of any effects cannot be predicted or
quantified at this time.

INFORMATION SYSTEMS FOR THE YEAR 2000

     The inability of some computer programs and embedded computer chips to
distinguish between the year 1900 and the year 2000 (the "Year 2000 problem")
poses a serious threat of business disruption to any organization that utilized
computer technology and computer chip technology in their business systems or
equipment. In proactive response to the Year 2000 problem, PNR established a
"Year 2000" project to assess, to the extent possible, PNR's internal Year
2000 problem; to take remedial actions necessary to minimize the Year 2000 risk
exposure to PNR and significant third parties with whom it has data interchange;
and, to test its systems and processes once remedial actions have been taken.
PNR has contracted with IBM Global Services to perform the assessment and
remedial phases of its Year 2000 project.

     As of March 31, 1999, PNR estimates that the assessment phase is
approximately 99% complete on a worldwide basis and has included, among other
procedures, (1) the identification of necessary remediation, upgrade and/or
replacement of existing information technology applications and systems; (2) the
assessment of non-information technology exposures, such as telecommunications
systems, security systems, elevators and process control equipment; (3) the
initiation of inquiry and dialogue with significant third party business
partners, customers and suppliers in an effort to understand and assess their
Year 2000 problems, readiness and potential impact on PNR and its Year 2000
problem; (4) the implementation of processes designed to reduce the risk of
reintroduction of Year 2000 problems into PNR's systems and business processes;
and, (5) the formulation of contingency plans for mission-critical information
technology systems.

     PNR expects to complete the assessment phase of its Year 2000 project by
the end of the second quarter of 1999 but is being delayed by limited responses
received on inquiries made of third party businesses.

                                       5
<PAGE>
     As of March 31, 1999, PNR estimates that the remedial phase is
approximately 79% complete, on a worldwide basis, subject to the continuing
results of the third party inquiry assessments and the testing phase. The
remedial phase has included the upgrade and/or replacement of certain
applications and hardware systems. The remediation of non-information technology
is expected to be completed during July 1999 PNR's Year 2000 remedial actions
have not significantly delayed other information technology projects or
upgrades. The testing phase of PNR's Year 2000 project is expected to be
completed by October 1999 and all other information technology systems and
non-information technology remediation by the end of the second quarter of 1999.
None of PNR's costs related to the Year 2000 are passed through to the Trust.

     A failure to remedy a critical Year 2000 problem could have a materially
adverse effect on PNR's results of operations and financial condition. The most
likely worst case scenario which may be encountered as a result of a Year 2000
problem could include information and non-information system failures, the
receipt or transmission of erroneous data, lost data or a combination of similar
problems of a magnitude to PNR that cannot be accurately assessed at this time.

     In the assessment phase of PNR's Year 2000 project, contingency plans are
being designed to mitigate the exposure to mission critical information
technology systems, such as oil and gas sales receipts; vendor and royalty cash
distributions; debt compliance; accounting; and, employee compensation. Such
contingency plans anticipate the extensive utilization of third-party data
processing services, personal computer applications and the substitution of
courier and mail services in place of electronic data interchange. Given the
uncertainties regarding the scope of the Year 2000 problem and the compliance of
significant third parties, there can be no assurance that contingency plans will
have anticipated all Year 2000 scenarios.

     The Trustee has developed and is implementing a program to prepare its
systems and applications for the Year 2000, including those used to render
services to the Trust. In that connection, the Trustee intends to have such
systems and applications capable of processing, on and after January 1, 2000,
date, and date-related data consistent with the functionality of such systems
and applications, without a material adverse effect upon its performance of
services as Trustee. Third parties that the Trust conducts business with could
be prone to Year 2000 problems that could not be assessed or detected by the
Trust. The Trust is contacting the major third parties to determine whether they
will be able to resolve, in a timely manner, any Year 2000 problems directly
affecting the Trust and to inform them of the Trust's internal assessment of its
Year 2000 review.

     Information above with respect to PNR is based upon information provided by
PNR to the Trustee for use in this Form 10-Q.

FINANCIAL REVIEW

     During the first quarter of 1999, the Trust had no distributable income, as
compared to $656,038, representing $.0091 per unit, in the first quarter of
1998. The decrease in distributable income in 1999 was due primarily to a result
of lower natural gas production and prices and the recovery of capital costs
associated with completion costs on the Brazos A-7 No. 5 well in the fourth
quarter of 1998. The per unit amounts of distributable income for the first
quarter of 1999 and 1998 were earned by month as follows:

<TABLE>
<CAPTION>
                                         1999       1998
                                       ---------  ---------
<S>                                    <C>        <C>
January..............................  $  --      $   .0035
February.............................     --          .0017
March................................     --          .0039
                                       ---------  ---------
                                       $  --      $   .0091
                                       =========  =========
</TABLE>

                                       6
<PAGE>
     There was no Royalty income in the first quarter of 1999 as compared to
$694,318 for the first quarter of 1998. The decrease in Royalty income is
primarily due to substantially lower production volumes on South Marsh Island
blocks 155 and 156 and West Delta 61 and 62 blocks and decreased prices for
natural gas and crude oil, condensate and natural gas liquids in 1999.

     Production volumes for natural gas decreased to 276,120 Mcf in the first
quarter of 1999 from 474,074 Mcf in the first quarter of 1998. The average price
received for natural gas was $1.71 per Mcf in the first quarter of 1999 compared
to $2.57 per Mcf in the first quarter of 1998.

     Crude oil, condensate and natural gas liquids production decreased to 5,791
barrels in the first quarter of 1999 from 9,083 barrels in the first quarter of
1998. The average price received for crude oil, condensate and natural gas
liquids was $10.93 per barrel in the first quarter of 1999, compared to $16.71
per barrel in the first quarter of 1998.

     The decrease in natural gas and crude oil, condensate and natural gas
liquids production in the first quarter of 1999 as compared to the first quarter
of 1998 was primarily attributable to natural production declines on the South
Marsh Island blocks 155 and 156 and West Delta 61 and 62 blocks.

OPERATIONAL REVIEW

     During the mid-1980's, PNR withheld approximately $3.5 million ($3.1
million net to the Trust) as a reserve for potential liabilities for royalty
claims made by the Mineral Management Service ("MMS"). The claims by the MMS
included, among other things, disputed transportation allowances attributable to
the Trust's South Marsh Island properties and payments received by PNR from
purchasers as settlements under gas purchase contracts. During 1998, PNR settled
all known claims with the MMS for $3.6 million ($3.2 million net to the Trust)
which significantly reduced the amount in the reserve. As of March 31, 1999, the
balance of the reserve, including accrued interest, was approximately $3.4
million ($3.1 million net to the Trust). Subsequent to March 31, 1999, PNR
determined that this reserve is no longer necessary. Thus approximately $3.1
million will be made available for release to the Trust, subject to recovery of
an approximate $1.0 million cost carryforward, and included, net of amounts
used to replenish the reserve for Trust expenses, in the May distribution.

     PNR has advised the Trust that during the first quarter of 1999 its
offshore gas production was marketed under short-term contracts at spot market
prices primarily to H&N, Limited. PNR has further advised the Trust 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 first quarter of
1999 were generally lower than spot market prices in the first quarter of 1998.

     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, industrial growth, conservation measures,
competition and other variables.

     The Brazos A-39 block experienced a decrease in natural gas production in
the first quarter of 1999 as compared to the first quarter of 1998 primarily due
to natural production decline. The Brazos A-7 block experienced a decrease in
production 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 drilled in the second quarter of
1997. During the fourth quarter of 1998, PNR incurred $7.35 million ($661,000
net to the Trust) of completion costs for the Brazos A-7 No. 5 well. At March
31, 1999, the cost carryforward resulting from the completion costs on the
Brazos A-7 No. 5 well, other capital expenditures and over distributions by PNR
was approximately $966,525. As such, beginning in the fourth quarter of 1998,
the Trust stopped receiving royalty income from PNR and no such royalty income
will resume until the $966,525 of cost carryforwards are recouped by PNR. In
addition, no distributions to

                                       7
<PAGE>
unitholders will be made until the Trust recovers administrative expenses paid
from the Trust's reserve fund during the period in which royalty income is not
paid to the Trust (approximately $215,978 at March 31, 1999). The No. 5 well
commenced production late in the fourth quarter of 1998 and is currently
producing at a rate of approximately 8 MMcf per day.

     The South Marsh Island 155 and 156 blocks experienced a decrease in
production in the first quarter of 1999 as compared to the first quarter of
1998, primarily due to the cessation of production from the A-19 well in late
1998 and natural production decline. This block is currently not producing, but
a workover was started in May 1999 to attempt to restore production. Results of
the workover should be available by the end of May 1999. PNR purchased 3-D
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 oil and in
natural gas production in the first quarter of 1999 as compared to the first
quarter of 1998 primarily due to the cessation of production of PNR operated
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% net profits
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 third quarter of 1998 which encountered
320 net feet of pay in eight Miocene sands below a true vertical depth of 7,500
feet. In the fourth quarter of 1998, the operator drilled one development well
and one exploratory well. The development well encountered 250 net feet of pay
in seven Miocene sands. The operator has elected to set a four-pile platform,
and production is expected in the second quarter of 1999. The exploratory well
tested a new fault block which was determined to be non-commercial. The
exploratory well was subsequently plugged in the first quarter of 1999. The
Trust will receive an 11.25% overriding royalty interest in these wells.

     Matagorda Island 624 oil and natural gas production decreased in the first
quarter of 1999 as compared to the first quarter of 1998, primarily due to
natural production decline. Gross producing rate of the block was approximately
1 MMcf of gas and 19 barrels of condensate per day as of May 1999.

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 1998 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, $128,000, $173,000 and
$123,000 for the years 1998, 1997, and 1996, respectively.

     The December 31, 1998 reserve report prepared for the Partnership indicates
that Royalty income expected to be received by the Trust in 1999 and 2000 could
be at or near the Termination Threshold. The

                                       8
<PAGE>
reserve report estimates that future Royalty income to the Trust is
approximately $5.2 million while the Termination Threshold for 1998 was
approximately $1.4 million. It is therefore possible (depending on the timing of
future production and drilling, market conditions, recoupment of unrecovered
capital costs, the receipt of amounts withheld from the Trust related to MMS
royalty claims, and other matters) that in either 1999 or 2000 Royalty income
received by the Trust may be below the Termination Threshold. If Royalty income
falls below the Termination Threshold for three successive years, the Trust
would terminate pursuant to the terms discussed above. There are numerous
uncertainties inherent in estimating and projecting the quantity and value of
proved reserves for the Trust properties as many of the Trust properties are
nearing the end of their productive lives and are therefore subject to
unforeseen changes in production rates. As such, there can be no assurance that
Royalty income received by the Trust in 1999 or 2000 will be above the
Termination Threshold.

     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.

                                       9
<PAGE>
     The following tables provide a summary of the calculations of the net
proceeds attributable to the Partnership's royalty interest (unaudited):

<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 MARCH 31, 1999:
    Ninety percent of gross
      proceeds.......................  $  401,395  $   73,813  $   (6,900)   $ 67,412    $   535,720
    Less ninety percent of --
      Operating expenditures.........    (114,138)   (166,169)   (177,757)    (49,415)      (507,479)
      Capital costs recovered........      (3,474)     --          --          (4,767)        (8,241)
      Accrual for future abandonment
         costs and interest on cost
         carryforward................     (11,727)     (1,500)     (5,848)       (925)       (20,000)
                                       ----------  ----------  ----------   ----------   -----------
    Net proceeds (excess costs)......  $  272,056  $  (93,856) $ (190,505)   $ 12,305    $   --
                                       ==========  ==========  ==========   ==========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                    $   --
                                                                                         ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................        (370)      5,610      --             551          5,791
                                       ==========  ==========  ==========   ==========   ===========
      Average sales price per Bbl....  $     8.48  $    10.74  $   --        $  11.24    $     10.93
                                       ==========  ==========  ==========   ==========   ===========
      Natural gas (Mcf)..............     219,783      27,440      (3,039)     31,936        276,120
                                       ==========  ==========  ==========   ==========   ===========
      Average sales price per Mcf....  $     1.84  $     0.49  $     2.27    $   1.92    $      1.71
                                       ==========  ==========  ==========   ==========   ===========
    Producing wells..................           4           2           3           1             10

THREE MONTHS ENDED MARCH 31, 1998:
    Ninety percent of gross
      proceeds.......................  $  301,821  $  325,239  $  350,120    $393,738    $ 1,370,918
    Less ninety percent of --
      Operating expenditures.........     (89,900)   (293,755)   (177,539)    (73,766)      (634,960)
      Capital costs recovered........      --          --          --         (21,571)       (21,571)
      Accrual for future abandonment
         costs.......................     (11,727)     (1,500)     (5,848)       (925)       (20,000)
                                       ----------  ----------  ----------   ----------   -----------
    Net proceeds.....................  $  200,194  $   29,984  $  166,733    $297,476    $   694,387
                                       ==========  ==========  ==========   ==========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                    $   694,318
                                                                                         ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................         280       6,494         594       1,715          9,083
                                       ==========  ==========  ==========   ==========   ===========
      Average sales price per Bbl....  $    14.59  $    16.96  $    15.51    $  16.52    $     16.71
                                       ==========  ==========  ==========   ==========   ===========
      Natural gas (Mcf)..............     116,525      84,424     128,905     144,220        474,074
                                       ==========  ==========  ==========   ==========   ===========
      Average sales price per Mcf....  $     2.56  $     2.55  $     2.64    $   2.53    $      2.57
                                       ==========  ==========  ==========   ==========   ===========
      Producing wells................           3           3           3           1             10
                                       ==========  ==========  ==========   ==========   ===========
</TABLE>

------------

o   The amounts shown are for Mesa Offshore Royalty Partnership.

o   The amounts for the three months ended March 31, 1999 and 1998 represent
    actual production for the periods November 1998 through January 1999 and
    November 1997 through January 1998, 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.

o   At March 31, 1999, the cost carryforward was $1.0 million of which $0.7
    million primarily related to well completion costs on the Brazos A-7 No. 5
    and other unrecovered capital costs and $0.3 million related to over
    distributions by Pioneer over the twelve months ending December 31, 1998.

                                       10
<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. Chase Bank of Texas,
National Association is the successor name of Texas Commerce Bank National
Association.)

<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.

                                       11
<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:  May 12, 1999

     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.

                                       12