The result was close for proposals to negotiate a permanent customs union with the EU.
This debate, this division, can not drag on much longer.
European Council President Donald Tusk gave a cautious welcome to May's change of course.
Earlier, EU negotiator Michel Barnier offered a downbeat assessment of the situation. "I have done everything I can to find a compromise".
Prime Minister Theresa May was to call in her cabinet to discuss the next steps after lawmakers failed to find a majority on any alternative to the divorce deal she struck with Brussels - an agreement they have also rejected three times already.
In her letter, obtained by the Guardian, the shadow foreign secretary set out her points ahead of an emergency shadow cabinet meeting which she was unable to attend.
May's words seemed to indicate that she was veering away from the possibility of a no-deal Brexit - but also that she has not given up on her own unloved withdrawal agreement.
She reiterated criticism of Mrs May and warned Mr Corbyn to be wary.
"I have failed, chiefly because my party fails to compromise", he said. It allows continued participation in the single market and a "comprehensive customs arrangement" with the European Union after Brexit - including a "UK say" on future European Union trade deals - would remain in place until the agreement of a wider trade deal which guarantees frictionless movement of goods and an open border in Ireland.
May's move infuriated pro-Brexit politicians, who say Britain must cut ties to the European Union in order to forge an independent economic policy.
May s divorce deal with the other 27 European Union nations has been rejected three times by parliament and patience is wearing thin in Brussels as the deadline to end Britain s 46-year membership nears with no agreement in sight.
Amid all the uncertainty, analysts have said the economic impact in Britain could be massive.
In a move that has caused some surprise among Conservatives, the Prime Minister said it was time for national unity to help resolve the stalemate over Brexit.
May and Corbyn met for two hours Wednesday, with both sides calling the talks "constructive". The cross-party bill was spearheaded by Labour's Yvette Cooper and the Conservative Oliver Letwin. under it, a legal mechanism is created where the Commons can instruct the Prime Minister to see Article 50 extension in absence of an approval resolution of Brexit withdrawal agreement.
"If the government won't act urgently, then parliament has a responsibility to try to ensure that happens even though we are right up against the deadline", she added.
Any revised deal would have to be agreed by Parliament by Wednesday, May 22 to avoid the United Kingdom having to stand in the European elections.
"We are now in a really unsafe situation with a serious and growing risk of no deal", Labour Party legislator Yvette Cooper said.
"No deal was never our desired nor intended scenario".