The poll is the first to put the EU exit camp in the lead.
Forty-three percent of respondents said they would vote to leave, 40 percent would vote to stay and 17 percent are still undecided, according to the poll by Survation research agency for the right-wing Mail on Sunday newspaper.
Read: UK government confirms EU referendum in Queen's Speech
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.
Survation's last poll published at the beginning of July showed 45 percent in favour of staying, 37 percent for leaving and 18 percent undecided.
Sunday's poll also showed that 22 percent 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.
Read: France says Britain EU referendum 'very risky'
British lawmakers in the House of Commons are expected on Monday to debate and vote on a draft bill to allow the referendum, which will then pass to the House of Lords for final approval.
Prime Minister David Cameron has said he will campaign to stay in the European Union only as long as he can negotiate a series of EU reforms intended to restore more sovereignty powers to Britain.
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