AMENDMENT NUMBER TWO TO THE AVIALL, INC.

Exhibit 10.13

AMENDMENT NUMBER TWO

TO THE AVIALL, INC.

AMENDED AND RESTATED 1998 DIRECTORS STOCK PLAN

THIS AMENDMENT TO THE AVIALL, INC. AMENDED AND RESTATED 1998 DIRECTORS STOCK PLAN (this “Amendment”), dated as of June 26, 2003, is made and entered into by Aviall, Inc., a Delaware corporation (the “Company”). Terms used in this Amendment with initial capital letters that are not otherwise defined herein shall have the meanings ascribed to such terms in the Aviall, Inc. Amended and Restated 1998 Directors Stock Plan (the “Plan”).

RECITALS

WHEREAS, Section XI of the Plan provides that the Board of Directors of the Company (the “Board”) may amend the Plan and the Board desires to obtain stockholder approval of the proposal to amend the Plan; and

WHEREAS, the Board submitted the proposal to amend the Plan to the Company’s stockholders at the 2003 Annual Meeting of Stockholders; and

WHEREAS, the Company’s stockholders approved the proposal to amend the Plan;

NOW, THEREFORE, in accordance with Section XI of the Plan, the Company hereby amends the Plan as follows:

1.    Paragraph C of Section IV of the Plan is hereby amended, effective as of January 1, 2004, by deleting the first sentence of said paragraph in its entirety and substituting in lieu thereof the following:

C.    Stock Grants—Each participating Director who has made an election to participate in the Plan pursuant to Paragraph B of this Section IV shall be eligible to receive annually, on the first New York Stock Exchange trading day in February of each calendar year (the “Grant Date”), following such election, in lieu of such Director’s annual retainer for service as a director of the Company (the “Annual Retainer”) a grant of Common Stock.

2.    Paragraph A of Section V of the Plan is hereby amended, effective as of January 1, 2004, by deleting said paragraph in its entirety and substituting in lieu thereof the following:

A.    Vesting—With respect to grants of Common Stock made prior to January 1, 2004, each grant of Common Stock made to a participating Director in accordance with Section IV shall be vested on the six-month anniversary of the Grant Date, so long as the Director has served continuously as a director of the Company during the intervening six-month period; provided, however, that a participating Director who leaves the Board of Directors of the Company following the completion of the term of service for which the Director was elected prior to the end of such six-month period or whose service during such six-month period was interrupted due to death or disability shall be vested in a pro rata number of such shares. With respect to grants of Common Stock made on or after January 1, 2004, each grant of Common Stock to a participating Director in accordance with Section IV shall be vested on the eleven-month anniversary of the Grant Date, so long as the Director has served continuously as a director of the

Company during the intervening eleven-month period; provided, however, that a participating Director who leaves the Board of Directors of the Company following the completion of the term of service for which the Director was elected prior to the end of such eleven-month period or whose service during such eleven-month period was interrupted due to death or disability shall be vested in a pro rata number of such shares. Except as described in the preceding sentence, in the event a Director’s service to the Company terminates before the shares have vested, then all shares granted to such Director which have not vested shall be canceled and such shares shall be forfeited and retransferred to the Company, with the Director having no further right or interest in such forfeited and retransferred shares. Notwithstanding anything contained in this Paragraph A to the contrary, in the event of a Change in Control, all shares previously granted to a Director pursuant to Paragraph C of Section IV shall immediately vest.

3.    Paragraph A of Section VI of the Plan is hereby amended, effective June 26, 2003 by deleting said Paragraph in its entirety and substituting in lieu thereof the following:

A.    The Board may, from time to time and upon such terms and conditions as it may determine, authorize the granting to an Eligible Director of rights to purchase shares of Common Stock upon the exercise of an option granted pursuant to this Section VI (“Option Rights”). Notwithstanding anything in this Section VI or elsewhere in this Plan to the contrary and subject to adjustment as provided in Section IX, no Director shall be granted Option Rights for more than 5,000 shares of Common Stock during any fiscal year of the Company.

4.    Subparagraph (ii) in Paragraph B of Section VI of the Plan is hereby amended, effective as of January 1, 2004, by deleting said subparagraph in its entirety and substituting in lieu thereof the following:

(ii)    With respect to grants of Common Stock made prior to January 1, 2004, each such Option Right shall become exercisable in full on such date or dates not less than six months nor more than five years from the Grant Date, so long as the Director has served continuously as a director of the Company during the intervening period, and shall become exercisable in full immediately in the event of a Change in Control, as defined in Section VII. With respect to grants of Common Stock made on or after January 1, 2004, each such Option Right shall become exercisable in full on such date or dates not less than eleven months nor more than five years from the Grant Date, so long as the Director has served continuously as a director of the Company during the intervening period, and shall become exercisable in full immediately in the event of a Change in Control, as defined in Section VII.

5.    Subparagraph (iii) in Paragraph B of Section VI of the Plan is hereby amended, effective as of January 1, 2004, by deleting said subparagraph in its entirety and substituting in lieu thereof the following:

(iii)    With respect to grants of Common Stock made prior to January 1, 2004, in the event of the termination of service on the Board by the holder of any such Option Rights other than by reason of disability or death, the then outstanding Option Rights of such holder may be exercised to the extent that they would be exercisable on the date that is six months and one day after the date of such termination and shall expire eleven months and one day after such termination, or on their stated expiration date, whichever occurs first. With respect to grants of Common Stock made on or after to January 1, 2004, in the event of the termination of service on the Board by the holder of any such Option Rights other than by reason of disability or death, the then outstanding Option Rights of such holder may be exercised to the extent that they would be exercisable on the date that is eleven months and one day after the date of such termination and shall expire eleven months and one day after such termination, or on their stated expiration date, whichever occurs first.

6.    Except as expressly amended by this Amendment and Amendment No. One to the Aviall, Inc. Amended and Restated 1998 Directors Stock Plan dated June 14, 2002, the Plan shall continue in full force and effect in accordance with the provisions thereof.

IN WITNESS WHEREOF, the Company has caused this Amendment to be duly executed as of the date first written above.