Exhibit 10.7


APOLLO STRATEGIC GROWTH CAPITAL
III

9 West 57th Street, 43rd Floor

New York, NY 10019

[____], 2022


APSG Sponsor III, L.P.

9 West 57th Street, 43rd Floor

New York, NY 10019


Re:       Administrative Services Agreement


 


Gentlemen:


This letter agreement by and between Apollo
Strategic Growth Capital III (the “Company”) and APSG Sponsor III, L.P. (“Sponsor”), dated as of the date
hereof, will confirm our agreement that, commencing on the date that securities of the Company are first listed on the New York
Stock Exchange (the “Listing Date”) and continuing until the earlier of the consummation by the Company of an initial
business combination or the Company’s liquidation (in each case as described in the Company’s Registration Statement
on Form S-1 (File No. 333-254835), as amended, filed with the Securities and Exchange Commission) (such earlier date hereinafter
referred to as the “Termination Date”):


(i) Sponsor shall make available, or cause
to be made available, to the Company, at 9 West 57th Street, 43rd Floor, New York, NY, 10019 (or any successor
location of Sponsor), certain office space, utilities, secretarial support and administrative services as may be reasonably requested
by the Company. In exchange therefor, the Company shall pay Sponsor the sum of $16,667 per month, for up to 24 months, commencing
on the Listing Date and continuing monthly thereafter until the Termination Date; and


(ii) Sponsor hereby irrevocably waives
any and all right, title, interest, causes of action and claims of any kind or nature as a result of, or arising out of, this letter
agreement (each, a “Claim”) in or to, and any and all right to seek payment of any amounts due to it out of, the trust
account established for the benefit of the public shareholders of the Company and into which substantially all of the proceeds
of the Company’s initial public offering will be deposited (the “Trust Account”), and hereby irrevocably waives
any Claim it may have in the future as a result of, or arising out of, this letter agreement, which Claim would reduce, encumber
or otherwise adversely affect the Trust Account or any monies or other assets in the Trust Account, and further agrees not to seek
recourse, reimbursement, payment or satisfaction of any Claim against the Trust Account or any monies or other assets in the Trust
Account for any reason whatsoever.


This letter agreement constitutes the entire
agreement and understanding of the parties hereto in respect of its subject matter and supersedes all prior understandings, agreements,
or representations by or among the parties hereto, written or oral, to the extent they relate in any way to the subject
matter hereof or the transactions contemplated hereby.




    	 



    	 



    




This letter agreement may not be amended,
modified or waived as to any particular provision, except by a written instrument executed by the parties hereto.


No party hereto may assign either this
letter agreement or any of its rights, interests, or obligations hereunder without the prior written approval of the other party;
provided that the Sponsor may assign this letter agreement or any of its rights, interests or obligations hereunder to an affiliate
without the prior written approval of the Company. Any purported assignment in violation of this paragraph shall be void and ineffectual
and shall not operate to transfer or assign any interest or title to the purported assignee.


This letter agreement, the entire relationship
of the parties hereto, and any litigation between the parties (whether grounded in contract, tort, statute, law or equity) shall
be governed by, construed in accordance with, and interpreted pursuant to the laws of the State of New York, without giving effect
to its choice of laws principles.


This letter agreement may be executed in
one or more counterparts, each of which shall for all purposes be deemed to be an original but all of which together shall constitute
one and the same letter agreement.


 


[Signature page follows]




    	 



    	 



    

 


	 	Very truly yours,
	 	 
	 	APOLLO STRATEGIC GROWTH CAPITAL III
	 	 
	 	By:	
	 	 	Name:	James Crossen
	 	 	Title:	Chief Financial Officer and Chief Accounting Officer


AGREED TO AND ACCEPTED BY:


APSG SPONSOR III, L.P.


 


	By:	 	 	 
	 	Name:	 	 
	 	Title: