UK Brexit Secretary David Davis, who is responsible for negotiating the UK's departure from the European Union, resigned on Sunday.
"David Davis has done the right thing, a principled and courageous decision", he said.
David Davis penned a deliberately caustic resignation letter last night outlining a "significant number" of disagreements with Number 10 over the previous year, from being obliged to accept the Brussels sequencing of negotiations, through to the compromises on the Northern Ireland border last December.
After the hours-long meeting at Chequers, May seemed to have persuaded the most vocal Brexit campaigners in the cabinet, including Davis, to back her plan to press for "a free trade area for goods" with the European Union and maintain close trade ties.
The prime minister will have a crunch meeting with members of her Tory party to discuss her plan in Parliament on Monday evening.
They accuse May of backtracking on her professed red lines in the negotiations.
There has been substantial disquiet among Brexiteers about the Chequers agreement proposals - shown to Germany's Angela Merkel before her own Cabinet, according to some reports - which would leave Britain effectively still inside the EU's Single Market for goods and agricultural products but outside the Single Market for services - such as it is - and subject to a "common rulebook" dictated by the European Union court, as well as bound to European Union standards on state aid, employment, and other regulations and forced to collect customs duties on the bloc's behalf, among other onerous obligations.
The double resignation on Sunday has the potential to derail May's government and set in motion a chain of events that could lead to an attempt to oust her as prime minister. Their decision may embolden pro-Brexit lawmakers to make a move against her.
Writing in The Daily Telegraph on Monday, Mr Rees-Mogg said: "If the proposals are as they now appear, I will vote against them and others may well do the same".
Some Brexit-supporting lawmakers are angry at the proposals, saying they will keep Britain tethered to the bloc and unable to change its rules to strike new trade deals around the world.
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He said he was anxious the government's negotiating approach would "lead to further demands for concessions" from Brussels.
The responsibility for leading the negotiations should now go to an "enthusiastic believer in your approach, and not merely a reluctant conscript", he said.
In a letter to Davis, May disagreed with his characterization of her plans, saying the deal she seeks "will undoubtedly mean the returning of powers from Brussels to the United Kingdom".
Arch-Brexiteer Jacob Rees-Mogg hit out at the "defeatism" in the Government's plans, warning that he would vote against them - and suggested other Eurosceptics may do the same.
UK Foreign Secretary Boris Johnson.
Johnson has not commented publicly since Friday.
In recent weeks, junior ministers Phillip Lee and Greg Hands have also resigned in order to vote against government - the former over Brexit, the latter over Heathrow.
Trade Secretary Liam Fox put his name to a newspaper article backing the plan, and Environment Secretary Michael Gove defended the agreement in a TV interview.
Another minister may be leaving soon as well, the BBC reported.
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