FIRST AMENDMENT TO THE 2004 EQUITY INCENTIVE PLAN

EXHIBIT 10.2

FIRST AMENDMENT TO THE

2004 EQUITY INCENTIVE PLAN

OF

APEX SILVER MINES LIMITED

1. The 2004 Equity Incentive Plan (the “Plan”) was approved by the Board of Directors in April 2004 and approved by shareholders effective May 27, 2004.

2. This First Amendment to the Plan (this “Amendment”) was approved by the Board of Directors in April 2008 and by shareholders effective May 28, 2008, in accordance with Section 12(a) of the Plan.

3. Section 2(n) of the Plan is hereby deleted and replaced in its entirety as follows:

(n) “Fair Market Value” means, as of any date, the value of the Ordinary Shares determined as follows:

(i) If the Ordinary Shares are listed on any established share exchange, or traded on the Nasdaq National Market or the Nasdaq SmallCap Market, the Fair Market Value of Ordinary Shares shall be the closing sales price for such share (or the closing bid, if no sales were reported) as quoted on such exchange or market (or the exchange or market with the greatest volume of trading in Ordinary Shares if such shares are traded on more than one such exchange or market) on the day of determination, as reported by such exchange or market or such other source as the Board reasonably deems reliable.

(ii) In the absence of such markets for the Ordinary Shares, the Fair Market Value shall be determined in good faith by the Board.

4. Section 4(a) of the Plan is hereby deleted and replaced in its entirety as follows:

(a) Share Reserve. Subject to the provisions of Section 11 relating to adjustments upon changes in Ordinary Shares, the Ordinary Shares that may be issued pursuant to this Plan shall not exceed in the aggregate Three Million (3,000,000) Ordinary Shares. Of this amount, the Two Million (2,000,000) Ordinary Shares approved by Shareholders effective May 27, 2004 (together with any of such Ordinary Shares that revert to the Plan as provided in Section 4(b), the “Original Share Reserve”), may be issued pursuant to any type of Share Award. The One Million (1,000,000) Ordinary Shares approved by Shareholders effective May 28, 2008 (together with any of such Ordinary Shares that revert to the Plan as provided in Section 4(b), the “Additional Share Reserve”) may be issued pursuant to a Share Award under Section 7(a). Any Share Award maybe issued from either the Original Share Reserve or the Additional Share Reserve in the Board’s discretion, to the extent Ordinary Shares remain available in such reserve. Share appreciation rights provided for in Section 7(b) hereof that are

payable only in cash will not reduce the number of Ordinary Shares available for Share Awards granted under the Plan.

5. Subsection 6(c)(i) of the Plan is hereby deleted and replaced in its entirety as follows:

(i) Exercise Price of a Nonqualified Share Option. The exercise price of each Nonqualified Share Option shall not be less than one hundred percent (100%) of the Fair Market Value of the Ordinary Shares subject to the Option on the date the Option is granted. Notwithstanding the foregoing, a Nonqualified Share Option may be granted with an exercise price lower than that set forth in the preceding sentence if such Option is granted pursuant to an assumption or substitution for another option in a manner satisfying the provisions of Section 424(a) of the Code.

6. Section 16 is hereby added to the Plan as follows:

16. Performance-Based Compensation under Section 162(m). Notwithstanding anything herein to the contrary, the performance criteria for any Share Award that is intended to satisfy the requirements for “performance-based compensation” under Section 162(m) of the Code shall be established by the Committee based on one or more Qualifying Performance Criteria selected by the Committee and specified in writing in accordance with the regulations pursuant to Section 162(m).

(a) Qualifying Performance Criteria. For purposes of this Plan, the term “Qualifying Performance Criteria” shall mean any one or more of the following performance criteria, applied to either the Company as a whole or to a business segment, subsidiary or Affiliate, and measured either annually or cumulatively over a period of years, on an absolute basis or relative to a pre-established target, to previous years’ results or to a designated comparison group, in each case as specified by the Committee in the applicable Option Agreement or Share Award Agreement: revenue; revenue growth; operating income (before or after taxes); pre- or after-tax income (before or after allocation of corporate overhead and bonus); earnings per share; return on equity; total shareholder return; return on assets or net assets; appreciation in and/or maintenance of the price of the Ordinary Shares or any other publicly-traded securities of the Company; gross profits; earnings (including earnings before taxes, earnings before interest and taxes or earnings before interest, taxes, depreciation and amortization); economic value-added models or equivalent metrics; comparisons with various stock market indices; reductions in costs; cash flow, cash flow per share or cash flow from operations; return on capital; improvement in or attainment of expense levels or working capital levels; operating margins, gross margins or cash margin; year-end cash; debt reductions; shareholder equity; regulatory achievements (including submitting or filing applications or other documents with regulatory authorities or receiving approval of any such applications or other documents); financing and other capital raising transactions

(including sales of the Company’s equity or debt securities); implementation, completion or attainment of objectives with respect to exploration, development, production or costs, acquisitions and divestitures, operational objectives, including those relating to environmental, health and safety requirements, recruiting and maintaining personnel, and joint venture or similar arrangements.

(b) Certification. Before payment of any compensation under a Share Award intended to qualify as “performance-based compensation” under Section 162(m) of the Code, the Committee shall certify the extent to which any Qualifying Performance Criteria and any other material terms under such Share Award have been satisfied (other than in cases where such relate solely to the price of Ordinary Shares).

(c) Discretionary Adjustments Pursuant to Section 162(m). Notwithstanding satisfaction or completion of any Qualifying Performance Criteria, to the extent specified at the time of grant of a Share Award to Covered Employees, the number of Shares or other benefits granted, issued, retained, or vested under a Share Award on account of satisfaction of such Qualifying Performance Criteria may be reduced by the Committee on the basis of such further considerations as the Committee in its sole discretion shall determine.

7. Except as provided in this Amendment, the Plan is unchanged and remains in full force and effect.