LONDON — Britain’s tottering efforts to leave the European Union remained at an impasse on Thursday as allies of Prime Minister Theresa May tried to keep her deal alive in the face of unbending opposition.

The deal, already defeated twice in Parliament, was given new life on Wednesday after Mrs. May promised Conservative Party colleagues who are hard-line supporters of leaving the bloc that she would step down if it were passed, and let someone else take over the next stage of talks with Brussels for the process known as Brexit.

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But even her offer to resign was not enough to assuage hard-line Conservatives or her putative allies in the Democratic Unionist Party of Northern Ireland, or D.U.P., who hate the deal.

So the government tried a new tack on Thursday: Ministers said they would disassemble Mrs. May’s deal into its two parts — one called the withdrawal agreement, and the other the political declaration — and ask Parliament to vote Friday only on the first.