Exhibit 5.1

Reed Smith LLP

599 Lexington Avenue
New York, NY 10022-7650

+1 212 521 5400

Fax +1 212 521 5450

reedsmith.com

September 13, 2024

Adagio Medical Holdings, Inc.

26051 Merit Circle, Suite 102

Laguna Hills, CA 92653

Ladies and Gentlemen:

We have acted as counsel to Adagio Medical Holdings, Inc. (f/k/a Aja Holdco, Inc.), a Delaware corporation (the “Company”), in connection with the preparation and filing by the Company with the Securities and Exchange Commission (the “Commission”) of a registration statement on Form S-1 (the “Registration Statement”) and the related prospectus filed with the Registration Statement (the “Prospectus”), covering the registration of up to (i) 7,905,387 shares of common stock, $0.0001 par value per share, of the Company (the “Common Stock”) issuable upon the conversion of those certain 13% senior secured convertible notes (the “Convertible Note Shares”) issued pursuant that certain securities purchase agreement, dated February 13, 2024, by and among the Company and certain investors named therein, and any assignment thereunder (the “Convertible Security Subscription Agreement”) and that certain note purchase agreement, dated February 13, 2024, by and among the Company, Adagio Medical, Inc., a Delaware corporation, and Perceptive Life Sciences Master Fund, Ltd (“Perceptive”), a Cayman Islands exempted company (the “2024 Bridge Financing Note Subscription Agreement” and together with the Convertible Security Subscription Agreement, the “Convert Subscription Agreements” ) and (ii) 600,000 shares of Common Stock (the “Convert Warrant Shares”) issuable upon exercise of warrants to purchase shares of Common Stock (the “Convert Warrants”) issued pursuant to the Convert Warrant Agreement, dated July 31, 2024 (including the form of warrant certificate included therein, the “Convert Warrant Agreement”) and the Convert Subscription Agreements.

This opinion letter is being furnished in accordance with the requirements of Item 601(b)(5) of Regulation S-K under the Securities Act.

We have reviewed originals or copies of (a) the Registration Statement, (b) the Business Combination Agreement, dated as of February 13, 2024, by and among the Company, Arya Sciences Acquisition Corp IV, a Cayman islands exempted Company (“ARYA”), Adagio Medical Inc., and the other parties thereto, as amended (the “Business Combination Agreement”), (c) the Company’s certificate of incorporation and bylaws, each as currently in effect, (d) the PIPE Subscription Agreements (e) the Convertible Security Subscription Agreement (f) the Convert Warrant Agreement (g) the Registration Rights Agreement, dated as of July 31, 2024, by and between the Company, Perceptive Life Sciences Master Fund, Ltd. and each holder thereto (the “Convert Registration Rights Agreement” and together with the Business Combination Agreement, the Convertible Security Subscription Agreement and the Convert Warrant Agreement, the “Transaction Documents”), and such other corporate records, agreements and documents of the Company, certificates or comparable documents of public officials and officers of the Company and have made such other investigations as we have deemed necessary as a basis for the opinions set forth below.

In rendering the opinion set forth below, we have assumed:

the genuineness of all signatures;

the legal capacity of natural persons;

the authenticity of all documents submitted to us as originals;

the conformity to original documents of all documents submitted to us as duplicates or conformed copies; and

as to matters of fact, the truthfulness of the representations and warranties made or to be made (as applicable) by the parties to the Transaction Documents and the truthfulness of the representations made in certificates or comparable documents of public officials and officers of the Company.

We have not independently established the validity of the foregoing assumptions.

Based upon the foregoing, and subject to the qualifications, assumptions and limitations stated herein, we are of the opinion that:

The Convert Warrant Shares have been duly authorized and, when issued and paid for upon exercise of the Convert Warrants in accordance with the terms of the Convert Warrant Agreement and the Convert Warrants, will be validly issued, fully paid and nonassessable.

The Convertible Note Shares have been duly authorized and, when issued and delivered upon conversion of the Convertible Notes in accordance with the terms of the Convertible Security Subscription Agreement, will be validly issued, fully paid and nonassessable.

The opinions stated herein are subject to the qualifications that we express no opinion as to the applicability of, compliance with, or effect of (i) public policy considerations which may limit the rights of parties to obtain certain remedies, (ii) any provision waiving the right to object to venue in any court; and (iii) any agreement to submit to the jurisdiction of any federal court.

Our opinion is limited to the General Corporation Law of the State of Delaware and the laws of the State of New York (the “Opined-on Law”) and we do not express any opinion herein concerning any other law. We advise you that issues addressed by this letter may be governed in whole or in part by other laws, but we express no opinion as to whether any relevant difference exists between the laws upon which our opinions are based and any other laws which may actually govern. This opinion letter speaks only as of its date.

We hereby consent to the filing of this opinion letter as Exhibit 5.1 to the Registration Statement and to the use of our name under the caption “Legal Matters” in the prospectus forming part of the Registration Statement and any amendments thereto. In giving such consent, we do not hereby admit that we are within the category of persons whose consent is required under Section 7 of the Securities Act, and the rules and regulations of the Commission promulgated thereunder.

	Very truly yours,

	/s/ REED SMITH LLP

	Reed Smith LLP