Mr Corbyn says the party will take this position to stop "no deal or a damaging Tory Brexit".
But he does not say what he would do if he won a general election and was placed in charge of the Brexit process.
Some senior members of his team want him to take a pro-Remain stance in all circumstances.
In an interview with the BBC's John Pienaar, Mr Corbyn said Labour was now the "party of choice" when it came to Brexit.
He said he had done "what I think a leader should do... an awful lot of listening" - to party members, unions and the wider Labour movement - before coming to a revised position.
He said he would "make a case" to Parliament in September to get another referendum and in the meantime, Labour will "do everything we can to take no deal off the table or stop a damaging deal of the sort Hunt or Johnson propose".
Asked if he had changed his position because of pressure from colleagues, Mr Corbyn said: "Not a bit of it. I've been listening and I've enjoyed it."
Mr Corbyn said he could not say what Labour's position would be at a general election, but would decide it "very quickly", depending on the circumstances at the time, whenever one was called.
In a letter to members, he said Labour continued to believe the "compromise plan" set out for Brexit during cross-party talks with the government earlier this year was a "sensible alternative that could bring the country together".
This included a customs union, a strong single market relationship and the protection of environmental regulations and rights at work.
- Can the next PM force through a no-deal Brexit?
- What is 'no deal'?
- How could another Brexit referendum work?
Mr Corbyn's statement followed a shadow cabinet meeting on Tuesday morning, and a meeting with trade union leaders on Monday.
The bosses of Labour's five-biggest affiliated unions called for the move the party has made - but also for it to hold a "confirmatory vote" on any new deal it negotiated if Labour won a general election.
'Bridge still to cross'
The BBC's political correspondent Iain Watson said there was disagreement about the second part of the unions' stance in shadow cabinet, with deputy leader Tom Watson wanting a "straight Remain stance", meaning a decision on it was "kicked down the road".
The deputy leader is among leading figures who have argued that confusion over Labour's message on Brexit contributed to its poor performance in the recent European Parliament elections.
Mr Watson said he was "happy" with the new Brexit position "up to the election", but the party had "yet to cross that bridge" when it comes to its manifesto for the next election.
"Our members have been telling us for some time now that they want us to be a Remain party and that they want us to put the new deal to the people," he added.
"We're now going to campaign for that and I'm very proud that the shadow cabinet have now listened to their concerns."
- Unions agree Brexit vote stance
- Labour must 'get on with' changing Brexit policy
- Don't panic over Brexit, McCluskey tells Labour
Shadow Treasury minister Clive Lewis said if a snap election was called, Labour would try to renegotiate the Brexit deal agreed by Theresa May, despite saying it "very much looks like" Labour is now the party of Remain.
He told the BBC's Politics Live: "If we win that general election, we will come into power, and if we can renegotiate that deal - a Labour deal - we will, because that's what people asked for."
Some Labour MPs, including Brexit-backing John Mann and Emma Lewell-Buck, said the party could lose votes in Leave-supporting areas as a result of the policy.
Ms Lewell-Buck, who quit her shadow minister role in opposition to a further referendum, said she was "concerned" and had a "heavy heart" over the decision.
"But I am also very clear that I am representing my constituents and I will continue to do that no matter what because they are the people who put me where I am," she said.
Analysis
GETTY IMAGES
By Nick Eardley, BBC political correspondent
There's always a "but", it seems, when it comes to Labour and Brexit. What the party is saying explicitly is that it'll try to force the new PM to hold another referendum and if that happens it will back Remain. But we don't know what Labour would do in the event of a general election.
The feeling among some Labour people is, "If you think it was hard to get here, trying to come to a manifesto position is going to be even harder," so they're just not going there yet.
Sometimes it feels Labour has been dragged kicking and screaming towards positions on Brexit, but it has at least got to a new one today. The question is whether it'll have to go further at some point.
Some on the Remain wing will be delighted with Jeremy Corbyn's shift, but others will feel there's more to do.
Former Labour MP Chris Leslie - who left the party to found Remain-backing Change UK - said Mr Corbyn's stance had "confirmed that if you vote Labour, you'll get Brexit".
He said the position "wasn't good enough", adding: "Brexit - whether a Labour Brexit or a Conservative Brexit - will cost people's jobs, put businesses in jeopardy, and diminish Britain in the eyes of our neighbours.
"Corbyn's refusal to be honest about that fact is a deep betrayal of the people Labour used to represent."
The Liberal Democrat's Brexit spokesperson, Tom Brake, said Labour "are still a party of Brexit".
He added: "Jeremy Corbyn can pretend all he likes that the Labour Party are finally moving towards backing the Liberal Democrat policy of a People's Vote, but it is clear it is still his intention to negotiate a damaging Brexit deal if he gets the keys to No 10."
But Miriam Mirwitch, chair of Young Labour, welcomed the move, adding: "This vital shift shows that Labour is a party centred around democracy that has listened to what it's members have wanted for some time: a People's Vote in which Labour campaigns to Remain."
Labour's position on Brexit
June 2017 - Labour's general election manifesto accepts referendum result
28 September 2018 - Labour agrees if a general election cannot be achieved it "must support all options… including campaigning for a public vote"
November 2018 - Shadow chancellor John McDonnell says Labour will "inevitably" back a second referendum if unable to secure general election
16 January 2019 - 71 Labour MPs say they support a public vote
6 February 2019 - Mr Corbyn writes a letter to Mrs May seeking five changes to her Brexit policy with no mention of a "People's Vote"
25 February 2019 - Labour says it will back a public vote if its proposed Brexit deal is rejected
14 March 2019 - Labour orders its MPs to abstain on an amendment calling for a second referendum
27 March 2019 - The party instructs its MPs to support Margaret Beckett's amendment which calls for a confirmatory public vote on any Brexit deal
30 April 2019 - NEC agrees that the European election manifesto will commit to a further referendum under certain circumstances
9 July 2019 - Labour calls on the next PM to hold a referendum and pledges to campaign for Remain against "no deal or a damaging Tory Brexit"
