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                                                                   EXHIBIT 10(b)

                            SECOND AMENDMENT TO THE
                      UNITED STATES LIME & MINERALS, INC.

                         EMPLOYEE STOCK OWNERSHIP PLAN

                 WHEREAS, United States Lime & Minerals, Inc. (the "Company")
         amended and restated the United States Lime & Minerals, Inc. Employee
         Stock Ownership Plan generally effective as of August 1, 1989 (the
         "Plan"); and

                 WHEREAS, the Company desires to liberalize participant
         diversification rights in order to permit participants who have
         attained age fifty (50) and completed ten (10) years of service in the
         Plan to transfer all or a portion of their Plan benefits into
         diversified investments maintained under an individual retirement
         account; and

                 WHEREAS, in response to the recent comments received from the
         Internal Revenue Service related to a pending determination letter
         request related to the Plan, the Company wishes to further amend the
         Plan to satisfy such comments.

                 NOW, THEREFORE, pursuant to the powers reserved in Section
         10.01 of the Plan, the Company does hereby amend the Plan, as follows:

                                       I.

Section 4.07(a) of the Plan is amended, effective August 1, 1996, to read as
follows:

         a.      Each Qualified Participant, as defined below, shall be
                 permitted to direct the Plan as to the investment of up to the
                 Applicable Percentage, defined below, of such Participant's
                 Accounts within one hundred and twenty (120) days after the
                 last day of each Plan Year during the Participant's Qualified
                 Election Period, as defined below. For the purposes of this
                 Section 4.07:

                          1.      "Qualified Participant" means a Participant
                                  who has attained age fifty (50) and who has
                                  completed at least ten (10) years of
                                  participation in the Plan.

                          2.      "Qualified Election Period" means the period
                                  beginning with the Plan Year in which the
                                  Participant becomes a Qualified Participant
                                  and ending with the Plan Year in which the
                                  balance to the credit of Participant's
                                  Account is distributed
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                          3.      "Applicable Percentage" means with respect to
                                  any Plan Year in the Qualified Election
                                  period, the following percentage:

<TABLE>
<CAPTION>
                                        Plan Year       Applicable Percentage
                                        ---------       ---------------------
                                       <S>                       <C>
                                            1                     25%
                                            2                     50%
                                            3                     75%
                                       4 and later               100%
</TABLE>

                                      II.

Section 4.08(e) of the Plan is amended, effective August 1, 1989, to read as
follows:

Repayments of principal and interest on any Acquisition Loan shall be made by
the Trustee (as directed by the Company or Committee) only from (I) Employer
Contributions, paid in cash to enable the repayment of such Acquisition Loan;
(2) Financed Shares or other collateral given for the loan in the event of
default or foreclosure; (3) earnings on such Employer Contribution and any
dividends or other earnings received by the Trust on such Financed Shares; (4)
earnings attributable to such permissible collateral other than Financed
Shares; and (5) the, proceeds of a subsequent Acquisition Loan incurred to
repay an existing Acquisition loan repayments. The committee shall instruct the
Trustee as to the priority and source of Acquisition Loan repayments.

                                      III.

Section 9.05 of the Plan is amended, effective August 1, 1989 to read as
follows:

         a.      a contribution to the Plan made by mistake of fact, if such
                 contribution (to the extent made by mistake of fact) is
                 returned to the Employer within one year after payment of such
                 contribution. While losses attributable to returned
                 contributions will reduce the amount to be returned, any
                 earnings attributable to returned contributions will not be
                 returned. Such returns will only be made with Qualifying
                 employer securities after other assets have been exhausted.
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                                      IV.

Except as amended by this instrument, the Plan, as previously amended and
restated, shall remain in full force and effect.

IN WITNESS WHEREOF, the Company has caused this instrument to be executed as of
this 26th day of December, 1996.

                                    UNITED STATES LIME &
                                    MINERALS, INC.

                                    By: /s/ TIMOTHY W. BYRNE
                                        ---------------------------------------
                                    Title: Sr. VP of Finance and Administration
                                           and Chief Financial Officer