The Attorney General's Brexit legal advice has been published in full following the Government's defeat in the House of Commons on Tuesday.
It warns the United Kingdom could be left in “protracted and repeated rounds of negotiations” over the Irish backstop.
It is likely to be seized upon by Tory critics of Theresa May's deal, who argue that the terms of a backstop designed to keep the Irish border open will deny the UK the power to withdraw from a customs union without agreement from Brussels.
Mr Cox found that the protocol setting out the terms of the backstop "does not provide for a mechanism that is likely to enable the UK lawfully to exit the UK-wide customs union without a subsequent agreement".
And he said that - despite assurances from both London and Brussels that it is intended to be temporary - the protocol would "endure indefinitely" under international law until another agreement takes its place.
His advice warned: "In the absence of a right of termination, there is a legal risk that the United Kingdom might become subject to protracted and repeating rounds of negotiations.
"This risk must be weighed against the political and economic imperative on both sides to reach an agreement that constitutes a politically stable and permanent basis for their future relationship.
"This is a political decision for the Government."
The letter, dated November 13, emerged just minutes before Theresa May faced MPs in a weekly session of Prime Minister's Questions ahead of the second day of a five-day Commons debate on her deal.
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During Prime Minister's Questions, the SNP's Ian Blackford accused Mrs May of "concealing the facts on her Brexit deal".
The party's Westminster leader mocked the PM for having lost three votes yesterday and being found in contempt of Parliament, saying: "We were promised strong and stable - what we've got is a Government in crisis."
The MP for Ross, Skye and Lochaber asked: "Is it time that the Prime Minister took responsibility for concealing the facts on her Brexit deal from members in this House and the public?
"Will she take responsibility?"
In response, Mrs May said the copy of the full legal advice released on Wednesday was "no different" to the statement the Attorney General made on Monday.
Read the legal advice
According to shadow Brexit secretary Sir Keir Starmer, the legal advice reveals "central weaknesses in the Government's deal".
Writing on Twitter, Sir Keir said: "Having reviewed the Attorney General's legal advice, it's obvious why this needed to be placed in the public domain.
Sir Keir Starmer discussing the release of the legal advice
"All week we have heard from Government ministers that releasing this information could harm the national interest. Nothing of the sort. All this advice reveals is the central weaknesses in the Government's deal.
"It is unthinkable that the Government tried to keep this information from Parliament - and indeed the public - before next week's vote."
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The release of the legal advice has resulted in further calls for a greater say in the final outcome of Brexit.
Former Foreign Secretary Boris Johnson told ITV News: "I am campaigning very strongly to change this deal, which I think is not in the interests of our country and not in the national interest and that is my job.
"I have a great sense of responsibility because I did help with the Vote Leave campaign, and I am very proud of it, but it really breaks my heart to see what has become of it - this isn't Brexit."
International Trade Secretary Liam Fox said the Remain majority in Parliament may attempt to "steal" Brexit from voters.
He told the Commons International Trade Committee: "I think that there is, as I have written recently, a real danger that the House of Commons, which has a natural Remain majority, may attempt to steal Brexit from the British people.
"Which I think would be a democratic affront."
Green MP Caroline Lucas tweeted sections of the legal advice, which she said suggested it was not received by the Cabinet until November 13 - the day MPs first voted for it to be released.
Ms Lucas highlighted concerns in the document that the protocol setting out the backstop arrangements for Ireland would "endure indefinitely".
According to the extracts, she said, a review mechanism in the Brexit deal "does not provide a unilateral route out of the backstop" and there is "a legal risk that UK could become stuck in protracted and repeating rounds of negotiations".