Exhibit 5.1

October 8, 2024

Adagio Medical Holdings, Inc.

26051 Merit Circle, Suite 102

Laguna Hills, California 92653

Ladies and Gentlemen:

We have acted as counsel to Adagio Medical Holdings, Inc., a Delaware corporation (the “Company”), in connection with the filing by the Company on the date hereof of a Registration Statement on Form S-8 (the “Registration Statement”) with the Securities and Exchange Commission (the “Commission”) covering the offer and sale of up to 8,268,329 shares (the “Shares”) of the Company’s common stock, $0.0001 par value per share (“Common Stock”), consisting of (i) 4,472,592 shares of Common Stock, issued or issuable pursuant to the Adagio Medical Holdings, Inc. 2024 Equity Incentive Plan (the “2024 EIP”), (ii) 3,354,444 shares of Common Stock issuable pursuant to the Adagio Medical Holdings, Inc. Inc. 2024 Key Employee Equity Incentive Plan ( the “2024 KEEIP”) and (iii) 441,293 shares of Common Stock issuable pursuant to the Adagio Medical Holdings, Inc. 2024 Employee Stock Purchase Plan (the “2024 ESPP” and, together with the 2024 EIP Plan and the 2024 KEEIP, the “Plans”).

This opinion is being furnished at the Company’s request in connection with the requirements of Item 601(b)(5) of Regulation S-K under the Securities Act of 1933, as amended (the “Securities Act”), and no opinion is expressed herein as to any matter pertaining to the contents of the Registration Statement or the prospectus forming a part thereof, other than as to the issuance of the Shares.

In rendering the opinion hereinafter expressed, we have made such legal and factual examinations and inquiries, including an examination of originals or copies certified or otherwise identified to our satisfaction as being true reproductions of originals, of all such documents, records, agreements and other instruments, including the Registration Statement, the Plans, the Amended and Restated Certificate of Incorporation of the Company, the Amended and Restated Bylaws of the Company, and corporate minutes of the Company as we have deemed necessary and appropriate for the purpose of this opinion. We have assumed that there are no agreements or understandings between or among the Company and any participants in the Plans that would expand, modify or otherwise affect the terms of the Plans or the respective rights or obligations of any participants thereunder. We have further assumed the genuineness of all signatures, the authenticity of all documents, certificates and records submitted to us as originals, the conformity to original documents, certificates and records of all documents, certificates and records submitted to us as copies, and the truthfulness of all statements of fact contained therein.

Based upon the foregoing, and having regard to legal considerations and other information that we deem relevant, we are of the opinion that, as of the date hereof, when the Shares shall have been duly registered on the books of the transfer agent and registrar therefor in the name or on behalf of the purchasers, and have been issued by the Company against payment therefor in the circumstances contemplated by the Plans and assuming that the individual issuances, grants or awards under the Plans are duly authorized by all necessary corporate action of the Company and duly issued, granted or awarded and exercised and paid for, for consideration at least equal to the par value thereof, in accordance with the requirements of law and the Plans (and the agreements and awards duly adopted thereunder and in accordance therewith), the offer and sale of the Shares shall have been duly authorized and, when and to the extent that the Shares are issued in accordance with the foregoing, such Shares will be legally issued, fully paid and non-assessable.

We express no opinion herein as to the laws of any state or jurisdiction other than the General Corporation Law of the State of Delaware.

We hereby consent to the filing of this opinion as an exhibit to the Registration Statement and the reference to this firm therein. In giving this consent, we do not thereby admit that we are within the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations of the Commission.

Very truly yours,

/s/ Reed Smith LLP

Reed Smith LLP