In a significant hardening of the position of Tory Eurosceptics, Mr Rees-Mogg made clear that he and his colleagues will vote against Mrs May's deal if she fails to "deliver what she has said she would".
Writing in The Telegraph, Mr Rees-Mogg compared Mrs May's position to that of Sir Robert Peel, the former Conservative Prime Minister forced to quit after his party revolted over the repeal of the Corn Laws.
It comes amid reports that the Prime Minister has told aides that she is prepared to stand and fight if Tory MPs force a vote to oust her, saying she will not be bullied out of office by hardened Eurosceptics.
Downing Street has reportedly produced a third model for handling customs after the UK leaves the EU.
Details of the new plan have not been revealed publicly, but ministers are expected to discuss it at Chequers on Friday.
At least four Cabinet ministers - Sajid Javid, Michael Gove, Jeremy Hunt and Gavin Williamson - are said to be "on manoeuvres" and laying the ground for future leadership bids in the event that Mrs May should be forced to quit over Brexit.
On Friday the Prime Minister will host a meeting of her full cabinet at Chequers, her rural retreat, in a bid to thrash out difference over Britain's post-Brexit trade and customs arrangements, ahead of the expected publication of a white paper next week.
Downing Street hopes it has now found its way out of a bind on customs, and a significant part of finding a solution to maintaining an open border with the Republic of Ireland, the BBC reports today.
Mr Rees-Mogg, who leads a 60-strong group of Tory Eurosceptic MPs, said that it is time for Mrs May and her Cabinet to decide whether to stand by her pledges on Brexit or reduce "a once proud country" to a "tremulous state that sees Brexit as mere damage limitation".
He said: "Theresa May must stand firm for what she herself has promised. One former Tory leader, Sir Robert Peel, decided to break his manifesto pledge and passed legislation with the majority of his party voting the other way. This left the Conservatives out of office for twenty-eight years.
"At least he did so for a policy that works. At Chequers the Prime Minister must stick to her righteous cause and deliver what she has said she would, she must use her undoubted grace to persevere."
The comparison to Sir Robert is particularly pointed, as the former Conservative Prime Minister was forced to resign in 1846 after his party revolted against his decision to repeal the Corn Laws. He was forced to rely on the Whigs and Radicals to repeal the laws, and resigned on the same day. He never held office again.
Mr Rees-Mogg warned that he and other Tory Eurosceptics will vote against the final Brexit deal if it crosses a series of red lines. They include enabling Brussels to impose laws "either directly or indirectly" on the UK after Brexit and inhibiting Britains ability to strike trade deals and control migration .
He also said that Theresa May's "idiotic" Customs Partnership plan, which Eurosceptic MPs fear could lead to a major climbdown over Brexit, must be ripped up.
"Any attempt by the EU to impose its laws and court on the UK, either directly or indirectly, must be rejected. Any EU agreement that restricts the country's ability to make trade agreements with other states, restricts our ability to control our migration policy makes us pay to trade or interferes with our fishing waters could not be accepted. Indeed MPs would vote against such propositions in Parliament."
His comments have drawn criticism from senior MPs. Sir Alan Duncan, a minister of state at the Foreign Office, accused Mr Rees-Mogg of "insolence" in "lecturing and threatening" the Prime Minister.
Mr Rees-Mogg said the Prime Minister made a "personal contract" with the British people when she declared that Britain will leave the single market and customs union.
He singled out Greg Clark, the Business Secretary who has suggested that the UK should join a single market for goods and financial services, for criticism.
He said: "The Prime Minister said, as soon as she took office, that 'Brexit means Brexit' and in the last election, in her personal contract with the British people, she declared that we would leave the single market and customs union.
"At Chequers this week the nation will see if her promises are kept of if the policy advocated by a former member of the SDP [Greg Clark] wins favour."
The Cabinet must agree to withhold the £39billion Brexit divorce bill unless there is a firm guarantee of a trade deal before Britain leaves the EU.
Mr Rees-Mogg said that the failure to secure such a guarantee is "something I would strongly oppose in any vote in the House of Commons".
He said that leaving the EU "into the purgatory of a perpetual transition" would be foolish. It comes after Mr Clark, the pro-European Business Secretary, and Andrea Leadsom, the Eurosceptic Leader of the Commons, both suggested yesterday that the post-Brexit transition period could be extended beyond December 2020.
He said that "above all" the Cabinet must be clear that it is prepared to accept no-deal. "It must maintain the clear negotiating line set out at the beginning: no deal is better than a bad deal and plenty of bad deals are on offer," he says.
The original article can be found at - https://www.telegraph.co.uk/politics/2018/07/01/theresa-may-risks-collap...