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The campaign, which was launched last week, has attracted support from Leavers and Remainers.
The odds on a second national vote have halved since the beginning of the year, bookies have revealed.
Those backing another referendum include senior figures from all the major parties, including former Tory cabinet minister Justine Greening, ex-prime minister Tony Blair, Liberal Democrat leader Vince Cable, and Green Party leader Caroline Lucas.
More than two thirds of students would also back a vote on the final Brexit deal, according to a new poll.
It comes as Theresa May’s efforts to bypass Brussels and deal directly with European Union nations have been rebuffed, appearing to push Britain further towards a no-deal scenario.
The prime minister was also warned on Friday that the European Parliament would veto her Chequers deal unless she comes up with a new plan for the Irish border.
Adding fuel to the fire was international trade secretary Liam Fox’s assertion on Friday that a no-deal Brexit would be better than prolonging talks with Brussels, a move he described as a “complete betrayal” of voters.
The Independent kicked off its campaign for a Final Say on whatever Brexit terms emerge, along with a petition, earlier this week.
Ex-cabinet minister Ms Greening said: “I believe that putting the final decision on Brexit in the hands of the British people is the only way to break parliament’s impasse, and the right way to decide our country’s future.
“That’s why The Independent’s campaign is so vital and has my support – anyone who feels the same should back it.”
Sir Vince said: “The Liberal Democrats have been fighting for the British people to have the final say on any Brexit deal, or worse still, no deal. This vote must include an option for an exit from Brexit.
“I congratulate The Independent for joining this campaign. It’s both a brave and correct move.”
Ms Lucas said The Independent’s voice was "already making a huge impact".
Labour MP David Lammy, who has called for a second referendum since 2016, said: “The Independent’s new push for a referendum on the final Brexit deal sits, with other similar campaigns, on the right side of history.
“Allowing the public to choose whether we remain in the EU – or indeed choose, in full knowledge, to become poorer, less sovereign and diplomatically neutered – is the only democratic solution to this political nightmare.”
Former Conservative attorney general Dominic Grieve, Labour’s Chuka Umunna, Lib Dem MP Layla Moran, campaigner Gina Miller and ex-Labour spin doctor Alastair Campbell have also backed the campaign.
Theresa May has urged voters not to worry about Brexit, despite her government setting out plans to stockpile food, blood and medicine in case it goes badly.
She said people should take “reassurance and comfort” from news of the plans, to be implemented if the UK crashes out of the EU without an agreement in March.
But as a no-deal Brexit began to look more likely, Mr Blair took to Twitter to give his support to the campaign.
He said: “33m+ people voted in the EU referendum wanting a better country, not to be poorer, not to put further pressure on our already stretched health service, not for job losses or to stockpile food and medicines. It should be your #Finalsay.”
Have you backed our campaign? Click here to sign our petition.
Regardless of how you voted in the Brexit referendum, you deserve a say on the final deal.
That’s why we’re launching the Final Say campaign, to win for the British people the right to make that all-important decision on the Brexit deal. Come what may in the months ahead, we maintain our commitment to our readers to retain balance and present many different points of view. But on this subject we believe a referendum on the final deal is right.
We believe there are three reasons:
1. The people, on both sides of the debate, are losing faith with the current process. From Theresa May to parliament as a whole, a chaotic approach has delivered infighting, resignations and party politics, but little progress on the key issues that people care about.
2. The people on both sides should have the chance to finish what they started. With reality coming into focus, it’s only natural to check this future is what people really want. After years of negotiations behind closed doors, the people must not be shut out of the final decision.
3. The people – on both sides – know so much more now than we did in 2016. The campaigns – again, on both sides – were flawed and clouded by misinformation. Many myths have been exposed. Now we must check there is support for the facts of Brexit.
A full account of The Independent’s view is available here.
In 2016, the people were given their say on the principle of whether to stay in the EU or leave. Now the facts are becoming clearer, it’s time for the people to have the final say on the real deal – before it’s too late.
Christian Broughton, Editor, The Independent
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