The leftie leader pledged to support moves for a "public vote to prevent a damaging Tory Brexit".
It is the first time Mr Corbyn has officially backed a referendum which could end up keeping us in the EU – after sitting on the fence for months.
The Labour boss has been under increasing pressure from pro-Remain MPs and activists to support a so-called "people's vote", especially after nine MPs left last week – with eight of them joining the Independent Group.
But Brexiteers blasted him today for backing a vote which would disregard the result of the 2016 referendum.
And the policy is still fudged – with no clarity on what form the referendum would actually take or when it would be.
Labour announced this evening that if their plan for an alternative Brexit deal is defeated in the Commons this week, the party will "put forward or support an amendment in favour of a public vote to prevent a damaging Tory Brexit".
Mr Corbyn told his MPs: “The Prime Minister is recklessly running down the clock, in an attempt to force MPs to choose between her botched deal and a disastrous No Deal. We cannot and will not accept.
"One way or another, we will do everything in our power to prevent No Deal and oppose a damaging Tory Brexit based on Theresa May’s overwhelmingly rejected deal.
“That’s why, in line with our conference policy, we are committed to also putting forward or supporting an amendment in favour of a public vote to prevent a damaging Tory Brexit being forced on the country.”
Labour will also endorse an amendment by Yvette Cooper due to be voted on this Wednesday which would rule out No Deal by giving the Commons the right to delay Brexit instead.
Mr Corbyn's U-turn was welcomed by Remain backers who claimed it could be a crucial moment in forcing a second referendum.
David Lammy, a supporter of the People's Vote campaign, said: "It is welcome news that the Labour party is now, at long last, signalling it will accept the principle of giving the public the final say on Brexit.
"Even if MPs can agree some sort of deal, it would be wrong to force it on the British people when we now know so many of the promises made in 2016 have been broken and any deal will be worse than the one we’ve already got inside the EU."
Critics slammed the move – saying Mr Corbyn was selling out Leave-supporting Labour voters.
Tory MP Simon Clarke said: "Labour have been shaping up to try to overturn the decision of millions of northern voters for some time. Those voters will, rightly, feel bitterly betrayed."
Bob Seely added: "Shameless #Corbyn who wanted Brexit now backs a second referendum to save his sorry leadership."
Labour sources suggested the party would not back a second referendum this week and would instead focus on building support for its alternative plan to keep Britain in the EU customs union.
But if that fails, as is almost certain, Mr Corbyn's team will then instruct MPs to vote for a referendum.
Does Corbyn's U-turn bring a second referendum closer?
TODAY'S U-turn is the first time that Jeremy Corbyn has promised to back efforts to get a second referendum.
But a so-called "people's vote" is still some way away – with several more steps needed before it could actually happen.
And Westminster insiders suspect Mr Corbyn's move is more to do with keeping his MPs happy than trying to overturn Brexit – given his lifelong suspicion of the EU.
On Wednesday, the Commons will vote on the way forward for Brexit.
One of the amendments debated will express support for Labour's alternative Brexit deal which would keep Britain in the EU customs union.
Only if that motion fails – as seems almost certain – will Mr Corbyn formally back a second referendum.
Theresa May has vowed to bring her deal back to Parliament by March 12, and if it passes then Britain WILL quit the EU.
But if it fails, MPs will get a chance to vote on the next steps – and that would be the moment for Labour to support a referendum.
Even if Labour, the SNP, Lib Dems and Independents win enough Tory support to get a "people's vote", it's not clear what form the referendum would take.
No one knows whether it would have two or three choices on the ballot paper – with the PM's deal, No Deal and Remain all potential outcomes.
And to give Britain enough time for a second referendum, the EU would have to allow a delay of several months to the ongoing Brexit process.
Mr Corbyn has previously resisted all moves to reverse Brexit despite pleas from pro-EU Labour activists.
Today's move comes after nine of his MPs quit last week, with eight forming the anti-Brexit Independent Group.
It could end up backfiring by panicking Tory Brexiteers into voting for Mrs May's deal because they're worried it's the only way to deliver Brexit at all.
The party leader has tried to play off both sides on Brexit since the 2016 referendum in a bid to avoid alienating both Leavers and Remainers.
Mr Corbyn draws his core support from pro-EU left-wingers – but millions of Labour voters in the North and Midlands supported Brexit.
In the autumn, Labour party conference voted to keep the option of a second referendum on the table if Mr Corbyn's efforts to secure a General Election failed.
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