The poll is the first to put the EU exit camp in the lead.
Forty-three per cent of respondents said they would vote to leave, 40 per cent would vote to stay and 17 per cent are still undecided, according to the poll by Survation research agency for the right-wing Mail on Sunday newspaper.
The survey, which was conducted online on September 3 and 4 and had 1,004 respondents, had a margin of error of two percentage points.
Today's poll also showed that 22 per cent of those who wanted Britain to stay said they could change their mind if the migrants crisis worsened.
Twenty-nine percent of respondents also told the pollsters that Britain should not accept any more Syrian refugees -- the highest response to a question on how many refugees Britain should take in.
British lawmakers in the House of Commons are expected tomorrow to debate and vote on a draft bill to allow the referendum, which will then pass to the House of Lords for final approval.
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