With her own Conservative Party deeply divided and the opposition Labour Party promising to vote the deal down, Mrs. May faces what most consider to be an almost impossible task.

But she is dogged, and, with Britain scheduled to leave the European Union on March 29, she appears to be relying on giving legislators a stark choice: her deal or a chaotic exit without any deal.

If she fails, what happens next would be anyone’s guess. Mrs. May could face a leadership challenge or be left to put her deal to a second vote in the British Parliament. There could be a push for a softer Brexit, new elections or a second referendum. Or Britain could lurch toward a no-deal Brexit, an outcome no one wants on either side of the Channel.

In an open letter to the nation, Mrs. May said she would campaign “heart and soul” to get the deal approved. Speaking in Brussels on Sunday, she added that “the British people don’t want to spend any more time arguing about Brexit,” and that her agreement would allow Britons “to come together again as a country.