She pledged to quit before the next stage of negotiations with the EU, but only if Tories who have so far refused to back her deal finally fall into line.
The “emotional” announcement at the private gathering of Conservative MPs means that if her deal is approved, it would be a new prime minister that would decide how to approach talks on the UK’s future relations with the EU.
It comes as Ms May desperately tries to gather enough votes from rebel Tories and the Northern Irish DUP, to try and pass her deal so that the UK can leave the EU on 22 May.
At the meeting of the Tory Backbench 1922 Committee on Wednesday Ms May said: “This has been a testing time for our country and our party. We’re nearly there. We’re almost ready to start a new chapter and build that brighter future. But before we can do that, we have to finish the job in hand.”
She added: “And I have heard very clearly the mood of the parliamentary party. I know there is a desire for a new approach – and new leadership – in the second phase of the Brexit negotiations – and I won’t stand in the way of that.
“I know some people are worried that if you vote for the withdrawal agreement, I will take that as a mandate to rush on into phase two without the debate we need to have. I won’t – I hear what you are saying. But we need to get the deal through and deliver Brexit.”
The prime minister said that she would leave earlier than intended “in order to secure a smooth and orderly Brexit” but did not give an exact date at this point.
However, reports already have already begun to emerge that if Ms May’s deal is passed, then a leadership contest would begin on 22 May – the day Britain’s EU membership would end.
The announcement led some in the meeting to pay tribute to the prime minister, such as ex-chief whip Patrick McLoughlin who said “what a wonderful job” she had done.
Eurosceptic MP Pauline Latham left the meeting confirming that she was now ready to vote for the Brexit deal negotiated by the prime minister.
She added: “She’s made the right decision, she’s read the mood of the party – which is a surprise”.
There was applause and banging of tables as Ms May left the meeting, with Scottish secretary David Mundell tweeting: “PM is driven by a sense of duty and public service. Her decision today reflects again her putting the national interest ahead of personal interest.
“She remains determined to secure our departure from the EU on an orderly basis and will have my full support in doing so.”
The government must still overcome opposition from speaker John Bercow who has vowed to block a new vote on her deal unless there are changes to the proposition put to the House of Commons.
The prime minister must also still secure the backing of the Northern Irish DUP, who only this week have voiced public opposition to the deal.
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