The ICM survey for The Sunday Telegraph shows Labour on 30 per cent, one point higher than last month's poll, despite the party squabbles.
The Conservatives are unchanged on 40 per cent and the Liberal Democrats down one point on 18 per cent.
Repeated in a general election, the figures would give the Conservatives a very small Commons majority of roughly 16 seats
The poll also reveals voters are divided on whether Labour would do better or worse in the general election, expected on May 6, if the party replaced Gordon Brown as leader.
Some 41 per cent said the party's fortunes would improve with a new leader while 35 per cent said they would worsen.
Overall, Conservative voters thought Labour would fare better if the party ditched Mr Brown while supporters of Labour and the Lib Dems though he was Labour's best bet.
David Cameron was more likely to be trusted by voters on three key issues than Mr Brown, scoring clear wins on the NHS (by 43 per cent to Brown's 35 per cent), schools (46 per cent to the prime minister's 34 per cent) and the economy (43 per cent to Brown's 36 per cent).
Exactly the same questions were asked by ICM in 2007, just as Mr Brown was preparing to take over from Tony Blair. Then Mr Cameron was narrowly ahead on the NHS and schools while Mr Brown has a large lead on the economy.
The majority of recent polls have put the Tories close to 40 per cent and Labour close to 30 per cent – just as all parties begin their final preparations for the election campaign.
A cabinet minister said: “It doesn’t wholly surprise me that we are up slightly – even at the end of such a dreadful week. The dominant thing in people’s minds this week has been the weather, not a lot of politicians arguing.”
Because electoral arithmetic is skewed in favour of Labour and against the Tories, Mr Cameron’s party effectively needs a 40 per cent share of the vote and at least a 10-point lead to be sure of winning a majority of seats in the Commons.
ICM Research interviewed a random sample of 1003 adults aged 18+ by telephone on 6-7 January 2010. Interviewed were conducted across the country and results have been weighted to the profile of adults. ICM is a member of the British Polling Council and abides by its rules.