Labour still has a slim chance of seeing off the Tories at the next general election if it can show that the economy is beginning to improve by polling day, a Guardian/ICM poll reveals today.

The Conservatives have maintained their 12-point lead over Labour, and David Cameron has extended his advantage as the most popular potential prime minister of the three main party leaders, the poll shows. But the survey offers a sliver of hope to Gordon Brown that Labour can become the largest party in parliament – if he can persuade wavering Tory and Liberal Democrat voters that he has started to reverse the recession in time for an election next year.

Two-thirds of the electorate, 65%, say clear signs of a recovery in the economy by the expected date of the next election, May 2010, would "make no difference" to their voting choice. But 9% of Tory voters and 22% of Liberal Democrat voters say they are "more likely to consider voting Labour" if the economy shows clear signs of recovery by then.

That suggests roughly 8% of those voters presently planning to vote Conservative or Lib Dem could be persuaded to switch to Labour. The findings will reinforce some Tory anxiety that their lead is "soft", and confirm cabinet ministers' view, expressed in the second part of the Guardian's series on Labour tomorrow, that all is not yet lost for the governing party.

Analysts believe a dead heat between Labour and the Tories on share of the national vote is likely to make Labour the largest party in the Commons and possibly give Brown a workable majority, but the scale of the mountain facing him remains formidable. The poll shows the Tory lead at 12 points for the third month in a row, with the Tories and Labour both unchanged on 42% and 30% respectively. The Lib Dems are up two on 20%, and others are on 8%, including 3% for nationalist parties and 2% for the Greens.

David Cameron's personal lead over Gordon Brown is even bigger: 45% of those polled say he would make the best prime minister of the three main party leaders, compared with Brown's 24%. Cameron's 21-point lead over Brown is an increase of 13 points since the Guardian and ICM posed the same question in April 2008. About one in six Labour voters, 17%, say Cameron would be a better prime minister than Brown.

The poll shows that voters are weary of Labour. ICM asked voters which of two opposing thoughts they were more likely to agree with. Just one in four voters, 25%, said that "continuity is important; stick with Labour" was closer to their view than "time for a change". More than two in three, 69%, plumped for "time for a change". Some 27% of Labour voters chose "time for a change".

While some voters are prepared to reconsider voting Labour if the economy improves, most do not expect it to. Most voters think the economy is more likely to deteriorate under a Labour government (53%) than a Conservative one (30%). Overall, voters believe Britain will come out of the recession more quickly under the Tories (44%) than under Labour (35%).

Those polled say big cuts in public services are more likely under the Tories (47%) than Labour (37%). But the state of the economy and the public mood suggests that this is not necessarily the kind of deciding issue that it was in the 2001 and 2005 elections. Conservative voters themselves think cuts are more likely under their preferred government than under a Labour one (48% to 40%). Conversely, while 46% say taxes are more likely to go up under a Labour government, a sizeable minority, 39%, think they are more likely to go up under the Tories.

There is overwhelming support for the view that the government should "take active steps to reduce the gap between rich and poor". Eighty-two per cent of those polled agreed with that view, and only 15% did not. This will encourage those on the left who believe that ministers can be bolder in reducing income disparities.

Disillusionment with Labour has spread to its own supporters. Sixty-six per cent of voters agree that Labour has "run out of steam, out of ideas", including 27% of Labour voters. Sixty-one per cent agree Labour does not deserve to win the next election, including 18% of Labour voters. Of all those polled, some 60% disagree with the view that Labour is taking the country in the right direction. And 54% disagree that "they are still a safer bet with the economy".

The chances of Labour receiving a drubbing at the EU elections in June appear high; 68% said that people they knew would vote primarily on domestic issues and 21% on mainly European issues.

• ICM interviewed a random sample of 1,004 adults aged 18+ by telephone between 13 and 15 March 2009. Interviews were conducted across the country and the results have been weighted to the profile of all adults. Percentages may not add to 100 because of rounding. ICM is a member of the British Polling Council and abides by its rules. Further information at www.icmresearch.co.uk.