"It is with great regret that I have had to conclude that as things stand there is still not sufficient support in the House to bring back the deal for a third meaningful vote", May said in a statement to Parliament on Monday.
Dame Margaret's amendment states that in the case of a No Deal scenario, seven days before the United Kingdom leaves the European Union the House of Commons should be recalled to consider a motion on whether MPs approve the move, in indicative votes, or back requesting an extension to Article 50, the legal mechanism for leaving the bloc.
In his resignation letter, he said he feared that a no-deal Brexit would be an "economic experiment" that a small minority of the Conservative party and United Kingdom would champion.
It seeks to change the rules of parliament on March 27 in order to provide time for lawmakers to debate and vote on alternative ways forward on Brexit, a process often referred to as "indicative votes".
Third Vote on May's Deal?
MPs are to have a series of indicative votes on the way forward for Brexit after passing an amendment to take control away from Prime Minister Theresa May.
The confrontation may come to a head at a cabinet session expected on Monday (today).
The withdrawal deal she has negotiated with the European Union has been overwhelmingly rejected in the Commons twice, and it remains unclear whether she will bring it back a third time next week after she wrote to MPs saying she would only do so if there was "sufficient support". The measure was proposed by Oliver Letwin, a member of Theresa May's Conservative party, and must be passed before it takes effect.
And, in any event, the European Union appears to be more prepared for that for which they can not prepare than Britain.
The amendment had been tabled by Conservative MP Sir Oliver Letwin and Labour MP Hilary Benn.
New Zealand PM Ardern orders top-level judicial inquiry into Christchurch mosque attacks
Last week, PM Jacinda Ardern moved swiftly to ban the military-style rifles used in the Christchurch attack. Royal commissions were reserved for the gravest of events and the shootings was one of those, she said.
The result "upends the balance between our democratic institutions and sets a unsafe, unpredictable precedent for the future".
He added: "The government has failed and this House must, and I believe will, succeed".
Opposition leader Jeremy Corbyn raised the prospect of putting any proposal supported by lawmakers back to the people.
"It is time for the PM to channel the spirit of Moses in Exodus, and say to Pharaoh in Brussels - LET MY PEOPLE GO", he wrote, leaving some room however for possibly supporting her deal.
MPs have already voted against a "no deal" Brexit but this remains the default position unless they agree an alternative.
With the March 29 Brexit day set nearly two years ago days away and the withdrawal agreement lacking Parliament's approval, European leaders agreed to a postponement last week to avoid a chaotic cliff-edge departure that would be disruptive for the world's biggest trading bloc and deeply damaging for Britain.
After Mrs May suggested there would not be a no-deal unless MPs agreed, Tory Leaver Crispin Blunt accused her of "the most shameful surrender by a British leader since Singapore in 1942". The European Commission put out a statement saying that a no-deal departure is "increasingly likely".
Anxious at the deepening political crisis in London, EU leaders last week agreed to postpone Brexit to avoid a potentially catastrophic "no deal" divorce on March 29, when 46 years of ties were formally scheduled to end.
They include dozens of rebels in her Conservative Party, some opposition Labour Party MPs and the Northern Irish Democratic Unionist Party (DUP), which props up her minority government but has rejected her deal so far.
The newspaper said it had spoken to 11 senior ministers who "confirmed that they wanted the prime minister to make way for someone else" and planned to confront her at Monday's cabinet.