The Opinium survey in The Observer newspaper put the Brexit Party of eurosceptic figurehead Nigel Farage on 34 percent, doubling its lead over the main opposition Labour Party to 13 points in a fortnight -- and on course to get more votes than Britain's two main parties put together.
Prime Minister Theresa May's governing Conservatives are in fourth place on 11 percent.
In the 2016 referendum on Britain's EU membership, 52 percent voted to leave while 48 percent voted to remain.
Adam Drummond, head of political polling at Opinium, said Remain voters were scattered among several parties.
"While 63 percent of Leavers say they will vote for the Brexit Party in the European elections, the most popular party among Remainers (Labour) only has 31 percent, versus 22 percent for the Lib Dems and 14 percent for the Greens," he said.
Brexit voters have deserted the Conservatives to such an extent that they now have a higher share of Remainers (12 percent) backing them than of Leavers (11 percent), he added.
In voting intentions for the next general election -- which must be held by May 2022 -- left-wing Labour is on 28 percent, the centre-right Conservatives on 22 percent, the Brexit Party on 21 percent and the centrist Liberal Democrats on 11 percent.
Opinium Research conducted an online survey of 2,004 British adults between Wednesday and Friday. The results were weighted.
The poll of 2,034 adults conducted on Thursday was commissioned by Brexit Express, a campaign group run by the Brexit Party's biggest donor.
May has pledged to step down once a Brexit divorce deal has been agreed upon by parliament.
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