SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549



                                  SCHEDULE 13G

                    Under the Securities Exchange Act of 1934



                            DORCHESTER MINERALS, L.P.
                                (Name of Issuer)

                                  Common Units
                         (Title of Class of Securities)


                                   25820R 10 5
                                 (CUSIP Number)


                                December 31, 2003
                              (Date of Event Which
                       Requires Filing of this Statement)



Check the appropriate box to designate the rule pursuant to which this Schedule
 is filed:

         [   ]  Rule 13d-1(b)

         [   ]    Rule 13d-1(c)

         [X]  Rule 13d-1(d)





     *The  remainder  of this cover  page  shall be filled  out for a  reporting
person's  initial  filing on this  form with  respect  to the  subject  class of
securities,  and for any subsequent amendment containing information which would
alter the disclosures provided in a prior cover page.

     The  information  required in the remainder of this cover page shall not be
deemed to be "filed" for the purpose of  Section 18 of the  Securities  Exchange
Act of 1934 ("Act") or otherwise  subject to the  liabilities of that section of
the Act but shall be subject to all other  provisions of the Act  (however,  see
the Notes).






Schedule 13G
CUSIP No. 25820R 10 5
1        NAME OF REPORTING PERSONS
         I.R.S. IDENTIFICATION NOS. OF ABOVE PERSONS (entities only)

         Preston A. Peak

2        CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP
         (See Instructions)
                   (a) G
                      --
                   (b) G
                      --

3        SEC USE ONLY

4        CITIZENSHIP OR PLACE OF ORGANIZATION

         United States citizen


NUMBER OF SHARES BENEFICIALLY OWNED BY EACH REPORTING PERSON WITH:
         5        SOLE VOTING POWER
                  1,577,412

         6        SHARED VOTING POWER
                  0

         7        SOLE DISPOSITIVE POWER
                  1,577,412

         8        SHARED DISPOSITIVE POWER
                  0

9        AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

         1,577,412

10       CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES
         (See Instructions)                                G
11       PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9
         5.8%

12       TYPE OF REPORTING PERSON (See Instructions)

         IN




Item 1.

         (a)      Name of issuer: Dorchester Minerals, L.P.

         (b)      Address of issuer's principal executive offices:
               3738 Oak Lawn Avenue, Suite 300, Dallas, Texas 75219


Item 2.

         (a)      Name of persons filing:  Preston A. Peak

         (b)      Address of principal business office:
               3738 Oak Lawn Avenue, Suite 300, Dallas, Texas 75219

         (c)      Citizenship:  United States

         (d)      Title of class of Securities:  Common Units

         (e)      CUSIP Number:  25820R 10 5


Item 3.  If this statement is filed pursuant to 240.13d(b) or 240.13d-2(b) or
 (c), check whether the person filing is a:

     (a)  [ ] Broker or dealer registered under section 15 of the Act (15 U.S.C.
          78o).

     (b)  [ ] Bank as defined in section 3(a)(6) of the Act (15 U.S.C. 78c).

     (c)  [ ]  Insurance  company as defined in section  3(a)(19) of the Act (15
          U.S.C. 78c).

     (d)  [ ] Investment  company  registered  under section 8 of the Investment
          Company Act of 1940 (15 U.S.C. 80a-8).

     (e)  [ ] An investment adviser in accordance with 240.13d-1(b)(1)(ii)(E);

     (f)  [ ] An employee  benefit plan or  endowment  fund in  accordance  with
          240.13d-1(b)(1)(ii)(F);

     (g)  [ ] A parent  holding  company or control  person in  accordance  with
          240.13d-1(b)(1)(ii)(G);

     (h)  [ ] A savings  association as defined in  Section 3(b)  of the Federal
          Deposit Insurance Act (12 U.S.C. 1813);

     (i)  [ ] A  church  plan  that  is  excluded  from  the  definition  of  an
          investment  company under section  3(c)(14) of the Investment  Company
          Act of 1940 (15 U.S.C. 80a-3);

     (j)  [ ] Group, in accordance with 240.13d-1(b)(1)(ii)(J).



Item 4.  Ownership

         (a)      Amount beneficially owned:  1,577,412

         (b)      Percent of Class:  5.8%

         (c)      Number of shares to which the person has:

                (i)  Sole power to vote or to direct  the vote:  1,577,412
                (ii) Shared  power to vote or direct the vote:  None
                (iii)Sole power to dispose or to direct the disposition  of:
                          1,577,412
                (iv) Shared  power to dispose or to direct the disposition of:
                           None


Item 5.  Ownership of Five Percent or Less of a Class

         Not applicable.


Item 6.  Ownership of More than Five Percent on Behalf of Another Person

         Not applicable.


Item 7.  Identification  and Classification of the Subsidiary Which Acquired the
     Security Being Reported on by the Parent Holding Company or Control Person

         Not applicable.


Item 8.  Identification and Classification of Members of the Group

         Not applicable.


Item 9.  Notice of Dissolution of Group

         Not applicable.

Item 10.  Certification

         Not applicable.


                              SIGNATURE

     After  reasonable  inquiry and to the best of my knowledge and belief,  the
undersigned  certifies that the information set forth in this statement is true,
complete and correct.



Dated:  February 17, 2004                           /s/ Preston A. Peak
                                                    --------------------
                                                        Preston A. Peak