"Let's bring the UK's withdrawal to an orderly end", the former Luxembourg premier told reporters, sitting next to May at a late-night press conference in the French city. "Because there will be no third chance", he said.
The three-part package of changes effectively aims to resolve a key sticking point for British MPs over the so-called backstop plan to keep open the border between European Union member Ireland and the British province of Northern Ireland.
If lawmakers vote down May's deal again, they will be given a vote on Wednesday on leaving without a deal, for which there is little support in parliament.
North East Herts MP Sir Oliver Heald also voted for the Prime Minister's deal. "Politicians must use any extension to work together as a matter of urgency to secure a deal, locking in a legally binding transition within the necessary timeframe". Another is if a delay is approved, what new compromises the European Union could even offer the UK.
But he added that if trade talks simply drag on because the two sides can not reach an agreement then Britain would not have a way out of the backstop.
So what happens on March 29?
The Daily Mail says there is now speculation over how long Mrs May will remain as Prime Minister, although her aides have insisted she has not considered resigning.
"I don't say it's going to happen, but clearly if a government can't get through on the one issue which we were really elected to deal with at the last election it puts us all in a very hard situation".
"Two weeks ago, I made a series of commitments from this despatch box regarding the steps we would take in the event that this House rejected the deal on offer".
The successful amendment was put forward by Yvette Cooper.
May's deal covers such things as citizens' rights, the status of the Irish border and Britain's divorce bill from the EU. Mrs May confirmed she would ask Brussels for an extension if ordered to by the Commons, although Downing Street said the Government would only seek a short delay of a few months.
The Commons later sent her back to renegotiate the backstop.
But Britain's attorney general, Geoffrey Cox, dealt a blow to May's plans, saying the assurances she had been given still meant the United kingdom could be locked in the bloc's orbit after Brexit, the most controversial issue for Brexit-supporting lawmakers.
Juncker said May and he have agreed a "legal instrument" to ease British concerns over the backstop.
"I am very, very suspicious and concerned about the time scale", Conservative lawmaker Andrew Bridgen said.
Tabled by Midlands MPs Caroline Spelman and Jack Dromey and backed by senior figures from all sides of the Commons including Sir Oliver Letwin, Hilary Benn, Nick Boles and Yvette Cooper, as well as all 11 members of the new Independent Group, this amendment simply rejects a no-deal Brexit at any time and under any circumstances.
Nigel Farage, who spearheaded the campaign that led to Britons voting in the June 2016 referendum to leave the European Union, held firm. Nothing has changed. Reject.
'A solution must come from London, Rutte said on Twitter, following Tuesday evening's no vote to the amended withdrawal deal thrashed out between Theresa May and European officials.