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                                                                     Exhibit 4.4

                FIRST AMENDMENT TO PROMISSORY NOTE, WARRANT AND
                         REGISTRATION RIGHTS AGREEMENT

     THIS FIRST AMENDMENT TO PROMISSORY NOTE AND WARRANT (the "Amendment") dated
as of July 18, 2000, is by and between Miller Exploration Company, a Delaware
corporation (the "Company"), and Veritas DGC Land, Inc., a Delaware corporation
("Veritas").

                              W I T N E S S E T H:

     WHEREAS, on April 15, 1999, the Company issued a promissory note (the
"Note") to Veritas in the principal amount of four million six hundred ninety-
six thousand forty and 60/100 dollars ($4,696,040.60), representing the
outstanding balance due Veritas for services provided to the Company in 1998 and
1999;

     WHEREAS, also on April 15, 1999, the Company issued a warrant (the
"Warrant") to Veritas that enables Veritas to purchase shares of the Company's
common stock, par value $0.01 per share (the "Common Stock"), in lieu of
receiving cash payments for the interest obligations under the Note, according
to the terms set forth in the Warrant;

     WHEREAS, also on April 15, 1999, the Company and Veritas entered into a
Registration Rights Agreement ("Registration Rights Agreement"), under which the
Company agreed to register shares of Common Stock issued upon exercise of the
Warrant as set forth therein;

     WHEREAS, as of the date hereof, the Warrant represents the right to
purchase 1,100,498 shares of Common Stock, which constitutes interest under the
Note from April 15, 1999 through October 14, 2000; and

     WHEREAS, the parties hereto desire to amend certain terms of the Note and
the Warrant.

     NOW, THEREFORE, in consideration of the premises and for good and valuable
consideration, the adequacy, receipt and sufficiency of which are hereby
acknowledged, the parties hereto hereby agree as follows:

     1.   Amendment to Note.
          -----------------

          (i) The Note is amended by deleting the introductory paragraph thereto
in its entirety and substituting in lieu thereof the following:

          FOR VALUE RECEIVED, MILLER EXPLORATION COMPANY, a Delaware
     corporation, promises to pay to the order of VERITAS DGC LAND, INC., a
     Delaware corporation, at Suite 932, 3701 Kirby Drive, Houston, Texas 77098-
     3982 (or such other place as the holder hereof may hereafter designate in
     writing), the principal sum of FOUR MILLION SIX HUNDRED NINETY-SIX THOUSAND
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     FORTY AND 60/100 DOLLARS ($4,696,040.60) (or the unpaid balance of all
     principal advanced against this note, if that amount is less), together
     with interest as follows: (a) provided that all obligations outstanding
     under this Note are paid in full by December 31, 2001 (the "Target Date")
     and subject to the provisions of Section 4 herein, at the Stated Rate, and
     (b) in the event that all obligations outstanding under this Note are not
     paid in full by the Target Date, at the Past Due Rate, subject to an
     additional adjustment as provided in Section 4 hereof; provided, that for
     the full term of this note the interest rate produced by the aggregate of
     all sums paid or agreed to be paid to the holder of this note for the use,
     forbearance or detention of the debt evidenced hereby (including, but not
     limited to, all interest on this note at the Stated Rate plus the
     Additional Interest) shall not exceed the Ceiling Rate.

          (ii) Section 1 of the Note is amended by deleting the following
definitions in their entirety and substituting in lieu thereof the following:

          "Credit Documents" means any and all papers now or hereafter
     governing, evidencing, guaranteeing, securing or otherwise relating to all
     or any part of the indebtedness evidenced by this note, including without
     limitation this note, the Warrant, the Registration Rights Agreement, and
     any and all existing and future amendments to such documents.

          "Maturity Date" means July 21, 2003.

          "Past Due Rate" means a rate per annum equal to the lesser of (i) the
     Ceiling Rate or (ii) thirteen and three-quarter percent (13 3/4%).

          "Stated Rate" means nine and three-quarters percent (9 3/4%).

          (iii)  Section 1 of the Note is also amended by adding the following
new definitions thereto:

          "Affiliate" means, as to any corporation, partnership, limited
     liability company, joint venture, association, joint-stock company, trust
     or unincorporated organization (an "Entity"), (a) any Entity controlling,
     controlled by or under common control with such Entity, (b) any director or
     officer of such Entity, (c) any Entity  that beneficially owns or holds
     five percent (5%) or more of the voting interest of such Entity, or (d) any
     natural person who, individually or with his immediate family members,
     beneficially owns or holds five percent (5%) or more of the voting interest
     of such Entity.

          "Amendment" means this Amendment to the Note, dated July 18, 2000.

          "Amended and Restated Credit Agreement" means the credit agreement
     entered into on or about July 18, 2000, by and between Maker and Bank One
     Texas, N.A, a national banking association, as such may be amended from
     time to time, a copy of which has been provided to Veritas.

                                       2
<PAGE>

          "Borrowing Base Compliance" means the compliance by Maker with the
     borrowing base requirements under the terms of the Amended and Restated
     Credit Agreement, as determined from time to time by the lender thereunder.

          "Change in Control" means (a) the sale, lease or other transfer of all
     or substantially all of the assets of Maker to any person or group (as such
     term is used in Section 13(d)(3) of the Securities Exchange Act of 1934, as
     amended); (b) the merger or consolidation of Maker with or into any other
     entity or the merger of another entity into Maker or any subsidiary thereof
     with the effect that immediately after such transaction the stockholders of
     Maker immediately prior to such transaction hold less than fifty percent
     (50%) of the total voting power of all securities generally entitled to
     vote in the election of directors, managers or trustees of the entity
     surviving such merger or consolidation; (c) the acquisition by any person
     or group, pursuant to a single transaction or series of related
     transactions, of more than fifty percent (50%) of the voting power of all
     securities of Maker generally entitled to vote in the election of directors
     of Maker (provided that this shall not be triggered by transactions with
     Guardian, Maker or Affiliates thereof); (d) a majority of the Board of
     Directors of Maker (the "Board") is composed of members who (i) have served
     less than twelve (12) months and (ii) were not approved by a majority of
     the Board at the time of their election or appointment; or (e) any
     transaction or series of related transactions, the result of which Maker is
     no longer subject to the reporting requirements of Section 13 of the
     Securities Exchange Act of 1934 or listed on Nasdaq.

          "Common Stock" means the Common Stock, par value $0.01 per share, of
     Maker.

          "Guardian" means Guardian Energy Management Corp., a Michigan
     corporation.

          "Guardian Transactions" means the transactions evidenced by that
     certain Securities Purchase Agreement, dated as of July 11, 2000, by and
     between Maker and Guardian, and the other documents related thereto, copies
     of which have been provided to Veritas.

          "Net Borrowing Base" means the engineered borrowing base, minus
     projected payments to Amerada Hess, plus projected cash flow after all
     payments (provided that the calculation of cash flow is satisfactory to the
     lender), minus capital expenditures.

          "Offer Term" means the period beginning on the date of this Amendment
     and ending on the first anniversary of the earlier of (i) the Maturity
     Date, or (ii) the date the obligations under the Note are paid in full, if
     prior to the Maturity Date.

          "Target Date" means December 31, 2001.

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          "Warrant Shares" mean shares of Common Stock that are issuable to
     Payee upon the exercise of the Warrant, which represent interest under the
     terms of this Note pursuant to Section 3 hereof.  The number of Warrant
     Shares, as may be increased from time to time, shall be determined in
     accordance with the terms of the Warrant.

          (iv) Section 3 of the Note is amended by reordering the paragraphs
thereof such that the current paragraph (c) becomes paragraph (d), and by adding
the following new paragraph (c) immediately preceding such paragraph (d) as
follows:

          (c) From and after October 15, 2000, interest hereunder shall continue
     to accrue, and shall be payable on April 15, 2001 and continuing on each
     October 15 and April 15 thereafter until the principal amount of the Note
     has been paid in full.

          (v) Section 3 of the Note is also amended by adding the following new
paragraphs (e), (f) and (g) at the end thereof:

          (e) Anything in this Note to the contrary notwithstanding and subject
     to paragraph (f) of this Section 3, payments of interest hereunder, if any,
     shall be paid exclusively by an increase in the number of Warrant Shares
     issuable upon the exercise of the Warrant, the number of which shall be
     determined in accordance with the terms of the Warrant; provided, however,
     that interest hereunder shall be paid exclusively in Warrant Shares only
     until such time as Maker is in Borrowing Base Compliance.

          (f) At such time as Maker has achieved Borrowing Base Compliance
     payments of interest shall be paid in cash on the date due; provided,
     however, that payments of interest shall be made only to the extent that
     such payments do not cause Maker's projected loan balance to exceed 80% of
     the Net Borrowing Base. In the event that Maker is unable to pay the full
     amount of any requisite interest payment in cash due to the restrictions
     set forth in the previous sentence, Maker shall pay as much of the interest
     payment in cash as possible, with the remainder of the interest to be paid
     in the form of additional Warrant Shares.  In connection herewith, Maker
     shall use commercially reasonable efforts to ensure that it achieves
     Borrowing Base Compliance and that it is able to pay interest due under
     this Note in cash as soon as practicable.  In furtherance of the foregoing,
     Maker shall promptly provide Payee with copies of all financial information
     and other reports that it is required to provide to its lender under the
     terms of the Amended and Restated Credit Agreement.  If Maker does not make
     a cash payment of interest on the date due, the interest shall be
     automatically made by an increase in Warrant Shares pursuant to Section
     3(e).

          (g) Maker may at any time pay the full amount or any part of this Note
     without payment of any premium or fee.

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     (vi) The Note is further amended by deleting the current Section 4 therein
and substituting in lieu thereof the following new Section 4:

          4.  Other Payments.  In addition to the other provisions of this Note
     regarding payment of the outstanding principal amounts hereunder, Maker
     covenants and agrees that:

          (a) Maker shall make a payment of five hundred thousand dollars
     ($500,000) to Payee on the effective date of the Amendment to be applied
     toward repayment of the outstanding principal under the Note.

          (b) Maker shall make an additional payment (the "Additional Principal
     Payment") of five hundred thousand dollars ($500,000) to Payee as set forth
     in this Section 4(b), which will be applied toward repayment of the
     outstanding principal under the Note. In order to fund this payment, Maker
     plans to sell approximately two million five hundred thousand dollars
     ($2,500,000) worth of Common Stock or Common Stock equivalents to Eagle
     Investments, Inc., in exchange for an equivalent amount of cash and
     property (the "Eagle Transaction").  The Board of Directors of Maker (i)
     has approved the Eagle Transaction and has recommended that the
     stockholders of Maker approve such transaction, (ii) will not rescind or
     revoke such recommendation, and (iii) will use its commercially reasonable
     efforts to ensure that the Eagle Transaction is consummated on or before
     October 31, 2000.  The Additional Principal Payment shall be paid to Payee
     within five (5) days after the Company obtains stockholder approval of the
     Eagle Transaction, provided, that if the stockholders fail to approve the
                        --------
     Eagle Transaction on or before October 31, 2000, then the Additional
     Principal Payment shall be paid on the earlier of (x) ninety (90) days
     following the date on which the Company fails to obtain such stockholder
     approval for, or the stockholders reject, the Eagle Transaction, or (y)
     December 31, 2000.

          (c) In the event that the payment in paragraph (b) is not made on the
     date or in the amount provided herein, the number of Warrant Shares
     issuable under the Warrant shall automatically be increased by an amount
     equal to Seven Hundred Fifty Thousand Dollars ($750,000), divided by the
     Market Value (as defined in the Warrant) of the Common Stock as of such
     date.  This increase in the number of Warrant Shares is a penalty, and is
     not in lieu of interest or principal due under the Note.

          (d) All proceeds derived from the exercise of the warrants issued to
     Guardian pursuant to the Guardian Transactions, and all proceeds of any
     kind whatsoever received by Maker pursuant to any transaction involving the
     issuance of equity or debt (other than borrowings under the Amended and
     Restated Credit Facility) subsequent to the Guardian Transactions, shall be
     paid only to Veritas until all amounts outstanding hereunder are paid in
     full, and shall be applied, to the extent consistent with the terms of this
     Note, first to the payment of interest and then to the payment of
     principal.

                                       5
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          (e) In the event that Maker does not pay Payee the $73,317.25 pursuant
     to Invoice No. 0700-LS-02 on or before August 4, 2000, the prevailing
     interest rate hereunder shall be increased by 400 basis points and shall
     apply retroactively to the date of the Amendment (subject to additional
     increases according to the terms hereof).

     (vii)  Section 6 of the Note is amended by adding the following new
paragraph (i) at the end thereof:

          (i)  A Change in Control of Maker.

     (viii)  The Note is further amended by adding the following new Sections 20
and 21 at the end thereof:

          Section 20.  Subordination.   Subject to the following sentence, all
     indebtedness and obligations of Maker to Payee evidenced by this Note,
     including without limitation, all obligations for the payment of principal,
     interest, penalties, fees, costs, expenses and indemnities, shall be and
     hereby are subordinated in right of payment and claim in favor of the
     indebtedness and obligations of Maker, not to exceed fourteen million
     United States dollars (US $14,000,000) in principal amount, under that
     certain credit agreement dated as of February 9, 1998, as amended from time
     to time, by and between Maker and Bank of Montreal, Houston Agency and any
     successor or replacement credit facilities or borrowings of Maker
     (including, without limitation, the Bank One Credit Agreement) that do not
     by their terms provide that they are subordinate to or pari passu with this
     Note (the "Senior Obligations").  Until the Maturity Date, Payee agrees
     that it will not request, demand, accept, or receive (by set-off or other
     manner), and Maker agrees that it will not pay or deliver to Veritas, any
     payment amount, credit, or reduction of all or any part of the amounts
     owing under this Note, provided, however, that nothing in this Section 20
     shall be interpreted to prohibit Maker from making, or Payee from receiving
     or otherwise exercising its respective rights to, the payments required
     pursuant to Sections 3, 4 and 21.

          Section 21.  Right of First Offer.  During the Offer Term, Maker
     shall, or shall cause its subsidiaries or contractors to, offer Veritas the
     right to perform any and all geophysical services with respect to
     properties of which Maker (or its subsidiaries) serve as the operator and
     has control over the selection of the geophysical services provider
     ("Geophysical Services").  In connection with this right of first offer,
     Maker shall notify Veritas in writing of any Geophysical Services to be
     performed during the Offer Term, and Veritas shall have fifteen (15) days
     within which to submit a bid in connection therewith before Maker contacts
     any third party providers (each, a "Third Party Provider"). In the event
     that Payee submits a bid to perform Geophysical Services, Maker (or its
     affiliate) shall negotiate in good faith with Payee with respect thereto.
     If, following such good faith negotiations, Maker (or its Affiliate) does
     not accept Veritas' bid and enters into a contract or

                                       6
<PAGE>

     subcontract for Geophysical Services with a Third Party Provider, Maker
     shall promptly deliver a copy of such contract or subcontract to Veritas to
     demonstrate that the terms thereof are materially superior to Veritas' bid.
     If, however, Veritas does not submit a bid to perform Geophysical Services
     as provided herein, Maker shall be free to enter into a contract or
     subcontract with any Third Party Provider without further notice to
     Veritas; provided, however, that this shall not limit or impair Veritas'
     rights with respect to any subsequent Geophysical Services to be performed
     during the Offer Term.

          Section 22.  Amendment to Credit Agreement.  Maker hereby agrees and
     covenants that it will not, without the prior written consent of Payee,
     agree to amend the Bank One Credit Agreement in a manner that could
     reasonably be expected to (i) adversely affect the rights of Payee under
     this Note, or (ii) extend the timing for payment of any amounts thereunder
     beyond the Target Date.

     2.   Amendment to Warrant.
          --------------------

          (i) The title of the Warrant is amended by deleting the caption
"Expiring on April 15, 2002," and replacing it with "Expiring on June 21, 2004,
subject to extension."  The introductory paragraph of the Warrant is amended by
deleting the final sentence thereof in its entirety and replacing it with the
following:

     This Warrant and all rights hereunder shall expire at 5:00 p.m., Houston,
     Texas time, on July 21, 2004, subject to extension as provided in Section
     2.1 hereof.

          (ii) Section 1 of the Warrant is amended by deleting the definitions
for the following terms in their entirety and substituting in lieu thereof the
following new definitions:

          "Note" shall mean that certain promissory note of the Company in the
           ----
     principal amount of $4,696,040.60, payable to Veritas and executed
     concurrently herewith, as the same shall be amended from time to time.

          "Warrant Shares" shall initially be an amount equal to nine percent
           --------------
     (9%) of the outstanding principal amount of the Note on April 15, 1999,
     divided by the Market Price.  On the six-month anniversary thereof, the
     number of Warrant Shares shall be increased by an amount equal to nine
     percent (9%) of the outstanding principal amount of the Note as of such
     date, divided by the Market Price.  In addition, on the one-year
     anniversary of the date thereof, the number of Warrant Shares shall be
     increased by an amount equal to nine percent (9%) of the outstanding
     principal amount of the Note as of such date, divided by the Market Value
     of the Common Stock as of such date. Beginning on April 15, 2001, and
     continuing on each April 15 and October 15 thereafter until all principal
     and accrued and unpaid interest due under the Note is paid in full, the
     number of Warrant Shares shall be increased by an amount equal to (A) the
     product of one-half of the prevailing rate of interest under the Note,
     multiplied by the principal amount outstanding thereunder, and (B) divided
     by the Market Value of the Common Stock as of such date; provided, that

                                       7
<PAGE>

     if the Company pays all or a portion of the Note during the interim, the
     number of Warrant Shares shall be increased in accordance with the formula
     set forth above, but shall be pro-rated to the date of payment.
     Notwithstanding the foregoing sentence, the Company shall have the right to
     reduce or eliminate all or any part of its obligation to increase the
     number of Warrant Shares on such respective dates by delivering to Veritas
     and/or its assigns all or a portion of the interest payment due such date
     in cash.

          (ii) Section 1 of the Warrant is amended by adding the following new
definition:

          "Amendment" means the amendment to the Warrant dated July 18, 2000.

          (iii)  Section 2.1 of the Warrant is hereby amended by deleting the
current Section 2.1 and substituting in lieu thereof the following Section 2.1

          2.1  Term; Exercise Price.  The Warrants represented hereby may be
               --------------------
     exercised by the holder hereof, in whole or in part, at any time and from
     time to time after the date hereof until 5:00 p.m., Houston, Texas time, on
     the earlier of (i) one year from the date the Note is paid in full, or (ii)
     July 21, 2004 ("Term"); provided, however, that to the extent that the Note
     is not paid in full by the Company on or before June 21, 2003, the Term
     shall be automatically extended on a daily basis so that the Term shall
     continue until the first anniversary of the date on which the Company has
     satisfied all of its obligations under the Note.  Subject to adjustment as
     provided in Article IV hereof, the exercise price for shares of Common
     Stock acquired under the Warrant shall be $0.01 per share ("Exercise
     Price").

     3.  Amendment to Registration Rights Agreement.
         ------------------------------------------

          (i) Section 2.1 of the Registration Rights Agreement is hereby amended
by deleting the current Section 2.1 and substituting in lieu thereof the
following Section 2.1:

          2.1  Registration.  Promptly after the date hereof, the Company shall
               ------------
     file a Form S-3 Registration Statement with the Securities and Exchange
     Commission to register shares of Common Stock for issuance upon exercise of
     the Warrant in an amount equal to the initial Warrant Shares (as defined in
     the Warrant), and shall use its reasonable best efforts to ensure that such
     Registration Statement is declared effective within 180 days of the date
     hereof.  The Company shall use its reasonable best efforts to file an
     additional such registration statement to cover any increase in the number
     of Registrable Securities underlying the Warrant, which registration
     statement shall be effective as promptly as practicable following the date
     of such increase. The Company shall also use its reasonable best efforts to
     keep such registration statements effective until at least six months after
     the expiration of the Warrant.  Each such registration statement referred
     to herein as a "Registration Statement."  The registration rights granted
     under this Section 2.1 are in addition to those provided in Section 2.2
     below, and the Company's obligations hereunder shall not terminate or be
     limited by the fact that piggy-back registration rights are available.

                                       8
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     4.   Counterparts.  This Amendment may be executed in any number of
          ------------
counterparts, each of which shall be considered an original for all purposes,
and all of which when taken together shall constitute a single counterpart
instrument.  Executed signature pages to any counterpart instrument may be
detached and affixed to a single counterpart, with such single counterpart with
multiple executed signature pages affixed thereto constituting the original
counterpart instrument.  All of those counterpart pages shall be read as though
one, and they shall have the same force and effect as if all the signers had
executed a single signature page.

     5.   Notice of Amendment; Indemnification.  Veritas hereby covenants and
          ------------------------------------
agrees that it will attach this Amendment to the Note and otherwise notify any
potential transferee of the Note of the existence of this Amendment.  Veritas
also covenants and agrees to indemnify the Company of any and all damages,
losses, claims, liabilities, demands, charges, suits, penalties, costs and
expenses (including court costs and reasonable legal fees and expenses incurred
in investigating and preparing for any litigation or proceeding) incurred by the
Company in the event that Veritas transfers the Note without giving the notice
required in the previous sentence.

     6.   Reaffirmation.  Except for the amendments to the Note and the Warrant
          -------------
expressly set forth herein, the Note, the Warrant and all other documents
executed in connection therewith shall remain in full force and effect, and the
Company and Veritas hereby restate and reaffirm all of the terms and provisions
of the Note, the Warrant and such other documents.  This Amendment shall not
affect the number of Warrant Shares currently issuable upon the exercise of the
Warrant or the exercise price in connection therewith.

     7.   Effective Date.  This Amendment shall be effective when executed by
          --------------
each party hereto as of the date first written above.

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<PAGE>

     IN WITNESS WHERE, the parties hereto have executed this Amendment as of the
date first written above.

                                    MILLER EXPLORATION COMPANY



                                    By:    /s/ Kelly E. Miller
                                        ----------------------------
                                        Name:  Kelly E. Miller
                                        Title:  President


                                    VERITAS DGC LAND, INC.



                                    By:   /s/ Deanna Goodwin
                                        ----------------------------
                                        Name:  Deanna Goodwin
                                        Title:  Vice President

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