<PAGE>   1
 
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                                   FORM 10-Q
 
                       SECURITIES AND EXCHANGE COMMISSION
 
                             Washington, D.C. 20549
 
(Mark One)
 
[X]          QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE
                      SECURITIES AND EXCHANGE ACT OF 1934
 
For the quarterly period ended June 30, 1998.
 
                                       OR
 
[ ]          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 1-8581
 
                   FREEPORT-MCMORAN OIL AND GAS ROYALTY TRUST
             (Exact name of registrant as specified in its charter)
 
<TABLE>
<S>                                                            <C>
                         TEXAS                                                72-6108468
     (State or other jurisdiction of incorporation                         (I.R.S. Employer
                    or organization)                                      Identification No.)
 
   CHASE BANK OF TEXAS, NATIONAL ASSOCIATION, TRUSTEE                            77002
                    712 MAIN STREET                                           (Zip Code)
                     HOUSTON, TEXAS
        (Address of principal executive offices)
</TABLE>
 
       Registrant's telephone number, including area code: (713) 216-5712
 
                      ....................................
 
   (Former name, former address and former fiscal year, if changed since last
                                    report)
 
     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  ______
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<PAGE>   2
 
                   FREEPORT-MCMORAN OIL AND GAS ROYALTY TRUST
 
                               TABLE OF CONTENTS
 
<TABLE>
<CAPTION>
                                                                PAGE
                                                              --------
<S>                                                           <C>
Part I.  Financial Information
 
  Item 1.  Financial Statements:
     Statements of Royalty Proceeds and Distributable
      Cash..................................................    3
     Statements of Assets, Liabilities and Trust Corpus.....    3
     Statements of Changes in Trust Corpus..................    3
     Notes to Financial Statements..........................    4
     Report of Independent Public Accountants...............    7
 
  Item 2.  Management's Discussion and Analysis of Financial
     Condition and Results of Operations....................    8
 
Part II.  Other Information
 
  Item 6.  Exhibits and Reports on Form 8-K.................    12
 
Signature...................................................    13
</TABLE>
 
                                      --2--
<PAGE>   3
 
                         PART I.  FINANCIAL INFORMATION
 
ITEM 1. FINANCIAL STATEMENTS.
 
                   FREEPORT-MCMORAN OIL AND GAS ROYALTY TRUST
             STATEMENTS OF ROYALTY PROCEEDS AND DISTRIBUTABLE CASH
                                  (UNAUDITED)
 
<TABLE>
<CAPTION>
                                            THREE MONTHS ENDED             SIX MONTHS ENDED
                                                 JUNE 30,                      JUNE 30,
                                        --------------------------    --------------------------
                                           1998           1997           1998           1997
                                        -----------    -----------    -----------    -----------
<S>                                     <C>            <C>            <C>            <C>
Royalty proceeds......................  $        --    $        --    $        --    $        --
Trust administrative expenses.........     (127,868)       (84,722)      (163,984)      (172,337)
Interest income.......................       18,865         19,416         38,229         38,227
Reserve for future Trust expenses.....      109,003         65,306        125,755        134,110
                                        -----------    -----------    -----------    -----------
Distributable cash....................  $        --    $        --    $        --    $        --
                                        ===========    ===========    ===========    ===========
Distributable cash per Unit...........  $        --    $        --    $        --    $        --
                                        ===========    ===========    ===========    ===========
Units outstanding.....................   14,975,390     14,975,390     14,975,390     14,975,390
                                        ===========    ===========    ===========    ===========
</TABLE>
 
               STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS
 
<TABLE>
<CAPTION>
                                                                JUNE 30,       DECEMBER 31,
                                                                  1998             1997
                           ASSETS                             -------------    -------------
<S>                                                           <C>              <C>
Cash and short-term investments.............................  $   1,579,826    $   1,705,582
Net overriding royalty interest in oil and gas properties...    189,875,741      189,875,741
Less, adjustment to recorded cost of net overriding royalty
  interest in oil and gas properties........................    (25,614,756)     (25,614,756)
Less, accumulated amortization of net overriding royalty
  interest..................................................   (164,260,985)    (164,260,985)
                                                              -------------    -------------
          Total assets......................................  $   1,579,826    $   1,705,582
                                                              =============    =============
                LIABILITIES AND TRUST CORPUS
Reserve for future Trust expenses...........................  $   1,579,826    $   1,705,582
Trust corpus (14,975,390 Units of Beneficial Interest
  authorized,
  issued and outstanding)...................................             --               --
                                                              -------------    -------------
          Total liabilities and trust corpus................  $   1,579,826    $   1,705,582
                                                              =============    =============
</TABLE>
 
                     STATEMENTS OF CHANGES IN TRUST CORPUS
                                  (UNAUDITED)
 
<TABLE>
<CAPTION>
                                                                SIX MONTHS ENDED
                                                                    JUNE 30,
                                                              --------------------
                                                                1998        1997
                                                              --------    --------
<S>                                                           <C>         <C>
Trust corpus, beginning of year.............................  $     --    $183,213
Royalty proceeds and interest income, net of trust
  administrative expenses and reserve for future Trust
  expenses..................................................        --          --
Distributions payable to Unit holders.......................        --          --
Amortization of net overriding royalty interest.............        --          --
                                                              --------    --------
Trust corpus, end of period.................................  $     --    $183,213
                                                              ========    ========
</TABLE>
 
   The accompanying notes are an integral part of these financial statements.
 
                                      --3--
<PAGE>   4
 
                   FREEPORT-MCMORAN OIL AND GAS ROYALTY TRUST
 
                         NOTES TO FINANCIAL STATEMENTS
1. THE TRUST
 
     Freeport-McMoRan Oil and Gas Royalty Trust (the Trust) was created
effective September 30, 1983. On that date, Freeport-McMoRan Inc. (FTX)
transferred a net overriding royalty interest in certain offshore oil and gas
properties to a Partnership (the Partnership) equal to 90 percent of the Net
Proceeds (as defined in the Conveyance referred to below) from FTX's working
interests in such properties and conveyed a 99.9 percent general partnership
interest in the Partnership to the Trust. Such net overriding royalty interest
is referred to herein as the "Royalty." The Overriding Royalty Conveyance which
created the Royalty is referred to herein as the "Conveyance." The Trust is
passive, with Chase Bank of Texas National Association as Trustee. The Trustee
has only such powers as are necessary for the collection and distribution of
revenues attributable to the Royalty, the payment of Trust liabilities and the
protection of Trust assets.
 
     The Trust Indenture provides generally that the Trust shall terminate upon
the first to occur of (i) the sale of all the Trust's interest in the
Partnership, or the sale by the Partnership of all the assets of the Partnership
including the Royalty, or (ii) a decision to terminate the Trust by the
affirmative vote of Unit holders representing a majority of the Units. The
Trustee is required to sell the Trust's interest in the Partnership, or cause
the Partnership to sell the Royalty, if the Trust's cash receipts for each of
three successive years are less than $3 million, thereby terminating the Trust
pursuant to (i) above. Upon the termination of the Trust under (ii) above, the
Trustee will sell the Trust's interest in the Partnership (or will cause the
Partnership to sell all of the assets of the Partnership). The Trustee will as
promptly as possible distribute the proceeds of any such sales according to the
respective interests and rights of the Unit holders after discharging all of the
liabilities of the Trust and, if necessary, setting up reserves in such amounts
as the Trustee in its discretion deems appropriate for contingent liabilities.
 
     The Trust's cash receipts last reached $3 million during 1995 and there
were no cash receipts in 1996 or 1997. Continued significant development of West
Cameron Block 498 has contributed to no cash being received during the first six
months of 1998. Additionally, the Class A cost carry-forward has increased to
$21.7 million at June 30, 1998 (Note 3). This cost carry-forward must be
recouped by the Working Interest Owner before any distribution may be made to
the Trust. Unless the Trust has cash receipts of at least $3 million during
1998, the Trust will be terminated by one of the means described above. The
actual level of cash receipts, if any, during the remainder of 1998 will depend
primarily on the timing and the rate of production from West Cameron Block 498,
the costs and results of future exploration and development costs on West
Cameron Block 498, oil and gas prices and other factors. Based on current
circumstances, it is unlikely that cash receipts to the Trust will total $3
million or more in 1998, in which case the Trustee would be required to sell the
Trust's interest in the Partnership or cause the Partnership to sell the
Royalty.
 
     A trust unitholder has advised the Trustee that he intends to seek
unitholder approval to convert the Trust into a corporation. The proposed
transaction would exchange each unit of beneficial interest in the Trust for one
share of a newly-formed ("NEWCO") corporation's common stock. The proposed
transaction must be approved by the Trust's Unitholders at a special meeting and
Newco common stock to be issued would have to
 
                                      --4--
<PAGE>   5
                  NOTES TO FINANCIAL STATEMENTS -- (CONTINUED)
 
be registered with the SEC under the Securities Act of 1933. Through July 21,
1998, there have been no further developments concerning this proposed
transaction.
 
2. THE ROYALTY
 
     IMC Global Inc. (IGL) succeeded to FTX effective December 22, 1997,
following the merger of FTX into IGL. Accordingly, IGL is now the Working
Interest Owner and presently owns the oil and gas interests burdened by the
Royalty. The Conveyance provides that the owner of the interests burdened by the
Royalty will calculate and pay monthly to the Partnership an amount equal to 90
percent of the net proceeds for the preceding month. Net proceeds generally
consist of the excess of gross revenues received from the Royalty Properties
(Gross Proceeds), on a cash basis, over operating costs, capital expenditures
and other charges, on an accrual basis (Net Proceeds).
 
3. DISTRIBUTIONS TO UNIT HOLDERS
 
     As a result of capital costs incurred in recent years, a cumulative excess
Class A cost carry-forward of $21,652,246 existed as of June 30, 1998. This
carry-forward is subject to and includes interest due the Working Interest Owner
at the prime rate, which totaled $812,689, net to the Trust during the six month
period of 1998. This excess Class A cost carry-forward must be recouped out of
future Net Proceeds before distributions to the Unitholders can be resumed.
 
4. GAS BALANCING ARRANGEMENTS
 
     As a result of past curtailments in gas taken by the former principal
purchaser of production from the Royalty Properties, certain quantities of gas
have been sold by other parties with interests in the Royalty Properties
pursuant to gas balancing arrangements. Proceeds from gas produced from the
Royalty Properties but sold by other parties pursuant to such balancing
arrangements (underproduction) are not included in Gross Proceeds for purposes
of calculating the Royalty. In the future, the Working Interest Owner will be
entitled to sell volumes equal to such underproduction or receive cash
settlements. On certain of the Royalty Properties, a cash settlement may be
required, depending on future results, because of the lack of sufficient
remaining reserves from which to makeup any underproduction. As of June 30,
1998, the unrecovered quantity of gas sold by third parties pursuant to such gas
balancing arrangements since inception of the Trust was approximately 1.4
billion cubic feet (bcf), net to the Trust. Gross Proceeds will be increased in
future periods when the Working Interest Owner is compensated either through the
sale of gas or through cash settlements, the amount and timing of which is
uncertain.
 
5. ESTABLISHMENT OF AN EXPENSE RESERVE
 
     Because of the decline in Royalty income, at certain times since late 1993
the Trust has been unable to pay its ongoing administrative expenses. To permit
the Trust to pay its routine administrative expenses during the time the Trust
incurs a Class A cost deficit, the Trustee, in accordance with the Trust
Indenture, established an expense reserve of $2.4 million, of which $1,579,826
remained as of June 30, 1998. Because of
 
                                      --5--
<PAGE>   6
                  NOTES TO FINANCIAL STATEMENTS -- (CONTINUED)
 
the cumulative excess Class A cost carry-forward (Note 3), $125,755 was
withdrawn from the expense reserve to pay Trust administrative expenses during
the first six months of 1998. There will be income tax consequences to the Unit
holders for such reserve as described in Note 6 below.
 
     The funding for this reserve is deposited with Chase Bank of Texas and
invested in Chase Bank of Texas collateralized certificates of deposit. The
average interest rate earned on these funds for the second quarter of 1998 was
4.6 percent.
 
6. FEDERAL INCOME TAX MATTERS
 
     Unit holders are required to report taxable income for the Royalty income
received by the Trust and deposited to the expense reserve even if no
distributions were received by the Unit holders. The expense reserve established
for Trust administrative expenses described in Note 5 above will give rise to
future income tax deductions as additional administrative expenses are incurred
and paid with funds deposited in the reserve.
 
7. RESERVE FOR FUTURE ABANDONMENT COSTS
 
     Estimated future abandonment costs are accrued over the life of the Trust's
properties based on current laws and regulations. During the 1997 fourth quarter
an updated assessment of estimated future abandonment costs was undertaken,
taking into consideration current labor and equipment costs levels and permitted
abandonment practices. This assessment resulted in a revision of estimated
remaining future abandonment costs to an amount that is approximately equal to
amounts previously withheld from distributions to Unitholders. Such costs are by
their nature imprecise and can be expected to be revised over time because of
changes in general and specific cost levels, government regulations, operation
or technology. As of December 31, 1997, the estimated remaining aggregate
abandonment costs to be incurred for all of the Trust's properties totaled $9.5
million net to the Trust, all of which has been withheld from distributions to
Unit holders. Any further adjustments to estimated abandonment costs or
variances to actual costs will reduce or increase future distributable cash
accordingly.
 
                             ---------------------
 
     THE INFORMATION FURNISHED HEREIN SHOULD BE READ IN CONJUNCTION WITH THE
FREEPORT-MCMORAN OIL AND GAS ROYALTY TRUST FINANCIAL STATEMENTS FOR THE YEAR
ENDED DECEMBER 31, 1997, WHICH ARE CONTAINED IN ITS ANNUAL REPORT ON FORM 10-K
FOR 1997.
 
                                      --6--
<PAGE>   7
 
                   FREEPORT-MCMORAN OIL AND GAS ROYALTY TRUST
 
                    REPORT OF INDEPENDENT PUBLIC ACCOUNTANTS
 
To Chase Bank of Texas National Association (Trustee)
and the Unit Holders of Freeport-McMoRan Oil and Gas Royalty Trust:
 
     We have reviewed the accompanying statement of assets, liabilities and
trust corpus of Freeport-McMoRan Oil and Gas Royalty Trust as of June 30, 1998,
the related statements of royalty proceeds and distributable cash for the three
month and six month periods ended June 30, 1998 and 1997 and the statement of
changes in trust corpus for the six month periods ended June 30, 1998 and 1997.
These financial statements are the responsibility of the Working Interest Owner.
 
     We conducted our reviews in accordance with standards established by the
American Institute of Certified Public Accountants. A review of interim
financial information consists principally of applying analytical review
procedures to financial data and making inquiries of persons responsible for
financial and accounting matters. It is substantially less in scope than an
audit in accordance with generally accepted auditing standards, the objective of
which is the expression of an opinion regarding the financial statements taken
as a whole. Accordingly, we do not express such an opinion.
 
     These financial statements were prepared on the cash basis of accounting,
which is a comprehensive basis of accounting other than generally accepted
accounting principles.
 
     Based on our reviews, we are not aware of any material modifications that
should be made to the financial statements referred to above for them to be in
conformity with the cash basis of accounting.
 
     We have previously audited, in accordance with generally accepted auditing
standards, the statement of assets, liabilities and trust corpus of
Freeport-McMoRan Oil and Gas Royalty Trust as of December 31, 1997 and, in our
report dated March 30, 1998, we expressed an unqualified opinion on that
statement.
 
                                          ARTHUR ANDERSEN LLP
 
New Orleans, Louisiana
July 21, 1998
 
                                      --7--
<PAGE>   8
 
ITEM 2.MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
       OF OPERATIONS.
 
     The information contained herein regarding operations and exploration and
development activities on the properties burdened by the Royalty has been
furnished by the Working Interest Owner who obtained this information from the
relevant operators. For convenience, references are made to the Notes to
Financial Statements.
 
     The following discussion includes forward-looking statements within the
meaning of Section 27A of the Securities Act of 1933 and Section 21E of the
Securities Exchange Act of 1934. Although it is believed that these
forward-looking statements are based on reasonable assumptions, important
factors could cause actual results to differ materially from those in the
forward-looking statements herein, including the timing and extent of changes in
commodity prices for oil and gas, environmental risks, drilling and operating
risks, risks related to exploration and development, uncertainties about Trust's
reserves, government regulations and other matters beyond the control of the
Trustee and the Working Interest Owner.
 
LIKELIHOOD OF TRUST TERMINATION
 
     As discussed in "Capital Resources and Liquidity" below and Notes 1, 3 and
10 to the financial statements included in the 1997 Annual Report on Form 10-K,
several factors occurring during 1997 increased the likelihood of the Trust
being terminated. The combination of a significant increase in the Class A cost
carryforward and negative reserve quantity revisions during 1997, combined with
declines in oil and gas prices received by the Working Interest Owner at
December 31, 1997 from those received at December 31, 1996, caused there to be
no proved oil and gas reserve quantities and related discounted future net cash
flows attributable to the Trust at December 31, 1997. Such proved reserve
estimates are based on various assumptions, many of which are subject to
uncertainties, as more fully discussed in the Trust's 1997 Annual Report on Form
10-K. These reserves estimates do not consider future changes in prices and
costs, the use of discount rates other than 10 percent or the possibility of
additional potentially recoverable reserves not currently classified as proved,
and therefore should not be considered to be a prediction of actual amounts to
be paid to the trustee or an estimate of fair market value. Further, as
discussed in Note 1, the Trust must have $3 million or more in cash receipts
during 1998 to avoid termination. Based on current circumstances, it is unlikely
that cash receipts to the Trust will total $3 million or more in 1998, in which
case the Trustee would be required to sell the Trust's interest in the
Partnership or cause the Partnership to sell the Royalty. On June 11, 1998, a
Trust unitholder announced his intention to seek unitholder approval to convert
the Trust's units into common stock of a newly-formed corporation. At this time
it is uncertain whether or not such proposal has the necessary support to
proceed. (See "Capital Resources and Liquidity."
 
OPERATIONAL ACTIVITIES
 
     Exploratory drilling on West Cameron Block 498 began in June 1994 and
ultimately resulted in four successful wells, which were saved for future
production. A 12 slot four pile drilling platform was set in March 1997, from
which four additional wells were drilled during the remainder of 1997 and early
1998. An auxiliary platform was set in October 1997, with production facilities
capable of handling 55 million cubic feet (Mmcf) of natural gas and 15,000
barrels of oil per day. In February 1998, Coastal Oil & Gas Corporation
(Coastal), the operator, announced intended future development plans for
additional drilling and construction activity in this block during the remainder
of 1998, including drilling and completing four additional wells and setting a
                                      --8--
<PAGE>   9
 
six-well satellite platform during the third-quarter. At June 30, 1998, West
Cameron Block 498 had eight development wells that contributed to production
during the second quarter of 1998. In June 1998, one of these wells was
temporarily shut in, re-completed and placed back on production in early July
1998. The B-9 well commenced during April 1998, was completed and logged in late
June and placed on production in early July 1998. The B-10 development well
commenced drilling in early July 1998 and was completed in early August 1998.
The drilling rig has been moved off location. The #6 exploratory well commenced
drilling during April 1998, was drilled and temporarily abandoned. A new
platform is currently being fabricated for developing the #6 well. The platform
will be set at a separate surface location from existing platforms in the field
during the third quarter. The average gross daily production relating to
revenues received by the Trust from this field during the second quarter was
approximately 8,100 barrels of oil and 37 Mmcf of gas. The average gross daily
production figures may vary throughout the life of the field. These variations
may be attributable to a number of factors, including the number of wells on
production at a given point in time, natural depletion of wells, production
techniques, and adjustments to flow rates in order to optimally produce the
related reserves. Total capital expenditures net to the Trust during the first
six months of 1998 were approximately $6.0 million for this block, in which the
Working Interest Owner owns a 23.1 percent working interest and a 19.2 net
revenue interest.
 
                                      --9--
<PAGE>   10
 
RESULTS OF OPERATIONS
 
     There were no cash distributions to the Unit holders during the first six
months of 1998 or 1997, primarily because of significant capital expenditures.
During the three months and six months ended June 30 1998, Total Costs exceeded
Gross Proceeds by approximately $1.3 and $4.7 million, respectively, primarily
because of the capital costs incurred to explore and develop West Cameron Block
498. As a result, the Class A cost carry-forward increased to $21.7 million
(Note 3) net to the Trust as of June 30, 1998. Because of the Class A cost
carry-forward, the Trust administrative expenses were paid from the expense
reserve during the 1998 and 1997 periods. The calculation of distributable cash
for each period follows:
 
<TABLE>
<CAPTION>
                                            THREE MONTHS ENDED             SIX MONTHS ENDED
                                                 JUNE 30,                      JUNE 30,
                                        --------------------------    --------------------------
                                           1998           1997           1998           1997
                                        -----------    -----------    -----------    -----------
<S>                                     <C>            <C>            <C>            <C>
Gross Proceeds(1).....................  $ 4,544,162    $   799,415    $ 5,308,443    $ 1,614,627
Total Costs(2)........................   (5,872,103)    (4,746,328)    (9,983,170)    (6,968,118)
Excess Class A cost (recovery) carry-
  forward, net(3).....................    1,327,941      3,946,913      4,674,727      5,353,491
                                        -----------    -----------    -----------    -----------
Net Proceeds..........................           --             --             --             --
Percentage attributable to Royalty....         90.0%          90.0%          90.0%          90.0%
                                        -----------    -----------    -----------    -----------
Amounts payable attributable to
  Royalty.............................           --             --             --             --
Percentage attributable to the
  Trust...............................         99.9%          99.9%          99.9%          99.9%
                                        -----------    -----------    -----------    -----------
Royalty Proceeds......................           --             --             --             --
Trust administrative expenses.........     (127,868)       (84,722)      (163,984)      (172,337)
                                        -----------    -----------    -----------    -----------
                                           (127,868)       (84,722)      (163,984)      (172,337)
Interest income.......................       18,865         19,416         38,229         38,227
Reserve for future Trust
  expenses(4).........................      109,003         65,306        125,755        134,110
                                        -----------    -----------    -----------    -----------
Distributable cash....................  $        --    $        --    $        --    $        --
                                        ===========    ===========    ===========    ===========
</TABLE>
 
---------------
 
(1) Represents amounts received by the Working Interest Owner during the three
    and six month periods ended in May of such year.
(2) Represents amounts accrued by the Working Interest Owner during the three
    and six month periods ended in May of such year. Includes accrued interest
    of $446,386 and $812,689 for the three and six months during 1998 as well as
    $110,873 and $169,346 for the three and six months during 1997.
(3) Represents Class A costs incurred in the applicable period that remained
    outstanding as of the end of such period.
(4) Represents withdrawals from the Trust administrative expense reserve during
    the respective periods.
 
                                     --10--
<PAGE>   11
 
     Gross Proceeds, which includes gas and oil revenues, are calculated based
on amounts received by the Working Interest Owner. Operating information
follows:
 
<TABLE>
<CAPTION>
                                                     SECOND QUARTER            SIX MONTHS
                                                   -------------------     -------------------
                                                     1998       1997         1998       1997
                                                   --------   --------     --------   --------
<S>                                                <C>        <C>          <C>        <C>
Natural Gas
  Revenues (in millions).........................   $  2.0     $  0.4       $  2.7     $  0.8
  Sales volumes (in million cubic feet)..........      891        161        1,140        298
  Average realization (per thousand cubic
     feet).......................................   $ 2.22     $ 2.82       $ 2.33     $ 2.82
Oil
  Revenues (in millions).........................   $  2.5     $  0.4       $  2.6     $  0.8
  Sales volumes (in thousands of barrels)........    168.8       16.3        173.1       34.5
  Average realization (per barrel)...............   $15.21     $21.20       $15.32     $22.41
</TABLE>
 
     Revenues and volumes for oil and gas during the six months and three months
ended June 30, 1998 increased substantially over the comparable periods during
1997 primarily reflecting the increased development and subsequent production at
West Cameron Block 498. Production at West Cameron Block 498 commenced during
the fourth quarter of 1997 from six wells and the number of producing wells has
increased to eight as of June 30, 1998. The increased revenues from West Cameron
Block 498 were offset in part by normal production declines at the Trust's other
remaining properties and by lower oil and natural gas prices, most significantly
during the second quarter of 1998 when oil prices fell significantly.
 
     Costs consist of the following (in millions):
 
<TABLE>
<CAPTION>
                                                           SECOND QUARTER             SIX MONTHS
                                                        --------------------     --------------------
                                                          1998        1997         1998        1997
                                                        --------    --------     --------    --------
<S>                                                     <C>         <C>          <C>         <C>
Lease operating expenses..............................    $0.9        $0.4        $ 1.3        $0.3
Exploration and development costs.....................     4.4         4.2          7.6         6.4
Interest, abandonment costs withheld and other........     0.6         0.1          1.1         0.2
                                                          ----        ----        -----        ----
                                                          $5.9        $4.7        $10.0        $6.9
                                                          ====        ====        =====        ====
</TABLE>
 
     Lease operating expenses for the six months and second quarter ended June
30, 1998 increased from the comparable periods during 1997 primarily because of
the increasing production costs associated with West Cameron Block 498. In
addition, the six months ended June 30, 1997 contains a first quarter
adjustment, affecting insurance, which resulted in an approximate $0.4 million
credit to lease operating expense during that period. Exploration and
development costs for the six months and second quarter of both 1998 and 1997
primarily reflects the drilling activity at West Cameron Block 498. The increase
in interest is the result of the Class A carry-forward escalating throughout
1997 and the first six months of 1998. Abandonment costs, which have been
accrued each period on the estimate of costs required to abandon the Trust's
properties, are now fully accrued based on current estimates. See further
discussion under "Capital Resources and Liquidity" and Note 7 to the Financial
Statements.
 
CAPITAL RESOURCES AND LIQUIDITY
 
     The Trustee may sell or dispose of the Trust's interest in the Partnership,
or permit the Partnership to sell or dispose of all or any part of the Royalty
only as authorized by a vote of the Unit holders, upon termination
 
                                     --11--
<PAGE>   12
 
of the Trust and in certain other limited circumstances. However, the Trust is
directed to effect such a sale (without such vote) if the Trust's cash receipts
for each of three successive years are less than $3 million. The Trustee must
distribute the net proceeds of such sale (after satisfaction of any outstanding
liabilities) to the Unit holders.
 
     The Trust's cash receipts last reached $3 million during 1995 and there
were no cash receipts in 1996 or 1997. Significant development costs, primarily
at West Cameron Block 498, resulted in no cash receipts during the first six
months of 1998. Additionally, the Class A cost carry-forward has increased to
$21.7 million at June 30, 1998 (Note 3). This cost carry-forward must be
recouped by the Working Interest Owner before any distribution may be made to
the Trust. Unless the Trust has cash receipts of at least $3 million during
1998, the Trust will be terminated by one of the means described above. The
actual level of cash receipts, if any, during the remainder of 1998 will depend
primarily on the timing and the rate of production from West Cameron Block 498,
the costs and results of future exploration and development costs on West
Cameron Block 498, oil and gas prices and other factors. Based on current
circumstances, it is unlikely that cash receipts to the Trust will total $3
million or more in 1998, in which case the Trustee would be required to sell the
Trust's interest in the Partnership or cause the Partnership to sell the
Royalty.
 
     A trust unitholder has advised the Trustee that he intends to seek
unitholder approval to convert the Trust into a corporation. The proposed
transaction would exchange each unit of beneficial interest in the Trust for one
share of a newly-formed ("NEWCO") corporation's common stock. The proposed
transaction must be approved by the Trust's Unitholders at a special meeting and
Newco common stock to be issued would have to be registered with the SEC under
the Securities Act of 1933. At the time of the filing of this report there have
been no further developments concerning this proposed transaction.
 
     Total committed exploration and development costs for the remainder of 1998
are presently estimated by the Working Interest Owner to be approximately $2.4
million net to the Trust, primarily for the development of West Cameron Block
498. This estimate is derived from cost estimates provided by the operators of
the Royalty Properties and may vary from actual costs depending on the success
of drilling, particular circumstances encountered during drilling and many other
factors outside the control of the operator. These expenditures can be expected
to reduce and could further delay resumption of distributions to Unit holders.
 
     Additional exploration may be proposed by the operators of certain other
Royalty Properties. After analyzing each proposal, the Working Interest Owner
will determine whether or not to participate in additional exploratory
operations.
 
     Estimated future abandonment costs, based on current laws and regulations,
are accrued over the life of the Trust's properties (Note 7). As of June 30,
1998, the estimated remaining aggregate abandonment costs to be incurred for all
of the Trust's properties totaled $9.5 million net to the Trust, of which all
has been withheld from distributions to Unit holders. Any adjustments to the
estimated abandonment costs or variances will reduce or increase future
distributable cash accordingly.
 
     At certain times since late 1993, the Trust has been unable to pay its
ongoing administrative expenses. To permit the Trust to pay its administrative
expenses during the time the Trust incurs a Class A cost deficit, the Trustee,
in accordance with the Trust Indenture, established a $2.4 million Trust
administrative expense reserve to pay such expenses (Note 5), of which $1.6
million remained at June 30, 1998.
 
                                     --12--
<PAGE>   13
 
                          PART II.  OTHER INFORMATION
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K.
 
  (a) Not applicable
 
  (b) Reports on Form 8-K:
 
      No reports on Form 8-K were filed by the registrant during the quarter for
      which this report is filed.
 
                                     --13--
<PAGE>   14
 
                                   SIGNATURE
 
     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.
 
<TABLE>
<S>                                                      <C>
                                                         FREEPORT-McMoRan OIL AND GAS
                                                         ROYALTY TRUST
 
                                                         By: Chase Bank of Texas, National
                                                         Association,
                                                               Trustee
 
                                                                      By: /s/ PETE FOSTER
                                                         ---------------------------------------------
                                                                          PETE FOSTER
                                                            SENIOR VICE PRESIDENT AND TRUST OFFICER
</TABLE>
 
Date: August 14, 1998
 
                                     --14--
<PAGE>   15
 
                               INDEX TO EXHIBITS
 
<TABLE>
<CAPTION>
        EXHIBIT
         NUMBER                                       DESCRIPTION
         ------                                       -----------
<C>                           <S>
           27                 Financial Data Schedule
</TABLE>