As May sealed the deal in Strasbourg with EU leaders, her de facto deputy, David Lidington, updated parliament in London on the plans.
"This evening in Strasbourg the prime minister... (has) secured legally binding changes that strengthen and improve the withdrawal agreement and the political declaration," he said.
He said this should be enough to persuade MPs to vote for the agreement on Tuesday, just 17 days before Britain is scheduled to leave the EU.
"Tomorrow there will be a fundamental choice -- to vote for the improved deal or to plunge this country into a political crisis," he said.
Three new documents have been agreed to run alongside the withdrawal agreement, which governs Britain's exit terms, and the political declaration on future trade terms.
First is a "joint legally binding instrument" on the withdrawal text, which addresses the deal's controversial "backstop" plan to keep open the border between EU member Ireland and the British province of Northern Ireland.
This "provides confirmation that the EU cannot try to trap the UK in the backstop indefinitely, and that doing so would be an explicit breach" of commitments on both sides, Lidington said.
The backstop would keep Britain in a customs union with the EU if and until another way -- such as a new trade deal -- could be found to avoid checks on the Irish border.
If the EU failed to try to find an alternative to the backstop, Lidington said Britain could use this joint instrument "as the basis for a formal dispute through independent arbitration", and ultimately get the backstop suspended.
The document also emphasises that both Britain and the EU want to find an alternative to the backstop by December 2020.
Finally, the document would put assurances EU leaders about the temporary nature of the backstop "onto a legally binding footing", Lidington said.
A second document would supplement the political declaration, and outline commitments from both sides on moving swiftly to this new relationship.
Thirdly, Britain has put forward a "unilateral declaration" with legal status in international law emphasising the temporary nature of the backstop, Lidington said.
This would clarify that Britain believes there is nothing in the Brexit deal to stop it seeking to exit the backstop if the EU fails to live up to its commitments to find a replacement.
Theresa May's de facto deputy, David Lidington said this should be enough to persuade MPs to vote for the agreement on Tuesday, just 17 days before Britain is scheduled to leave the EU.