The Ipsos/Mori poll shows that when undecided voters are excluded 53 per cent want the UK to quit the EU and 47 per cent want to stay.
The telephone survey of 1,257 adults across Britain was conducted for the Evening Standard.
According to the poll, immigration has overtaken the economy as the most important issue influencing how people intend to vote – a state of affairs that will be seen as a boost to Boris Johnson and the Leave campaign.
The survey reveals massive distrust of claims made by politicians on both sides, but especially the economic fears raised by David Cameron and George Osborne as they have argued against Brexit.
Only 17 per cent of people believe the Chancellor’s flagship claim that UK households will lose £4,300 and be permanently poorer after a Brexit, despite the analysis carrying the full weight of the Treasury. Some 70 per cent think it is a falsehood.
Almost half believe Boris Johnson is telling the truth when he says that Britain sends £350 million a week to the EU, even though it has been criticised as misleading by independent experts. The claim is believed by 47 per cent and disbelieved by 39 per cent.
More people think they will be better off in five years outside the EU than inside, despite Government warnings of lost jobs and lower incomes if Britain leaves.
Immigration is now the most critical issue, cited as very important to their vote by 33 per cent, up five points in a month, including just over half of leave supporters.
The economy has fallen to second place in the league table of important issues, down five points to 28 per cent.
A fifth of people say they might change their mind - which means the outcome is impossible to predict.
Overall the poll suggests a significant shift to the Leave camp in the past month, which coincides with the official Vote Leave campaign focusing more strongly on immigration.
Over half - 52 per cent - of people backing Leave list immigration as very important to their vote, while 41 per cent of remain backers cite the economy as very important.