Support has grown by two points since January to 36% and opposition has fallen by one point to 46%, while 18% were undecided, according to the Panelbase poll for the Sunday Times and Real Radio.
Among those whose minds are made up, the results show the pro-union Better Together campaign leads by 56% to 44%.
For pro-independence group Yes Scotland, the poll shows attention will have to be focussed on female voters.
A total 47% of men support independence compared with 25% of women, with the gap growing by seven points since January.
Opposition stands at 40% for men and 52% for women.
Ivor Knox, director of Panelbase, told the Sunday Times: "The gender gap appears to be greater than ever, particularly among younger voters. While an absolute majority (51%) of males under 55 favour independence, men over 55 and women of all ages remain opposed."
The results were published as First Minister Alex Salmond's SNP meets in Inverness for its spring conference.
In his key note speech yesterday, he pledged that independence could see parents benefit from a European style system of childcare with a far greater level of public subsidy.
"Support for independence is rising and support against independence is falling," Mr Salmond told the BBC's Andrew Marr Show today.
"That's the narrowest gap we've had in the campaigning period so far and we've still got 18 months to go.
"The Yes campaign believes that if we can win the arguments, we'll win this referendum."
He revealed on Thursday that the referendum will be held on September 18 next year.
The Panelbase poll of 885 people, conducted over the past week, also shows strong support for the SNP at Holyrood.
Asked which party people would back in a Scottish election, 42% said SNP and 27% said Labour.
Annabelle Ewing, an SNP member of Holyrood's Referendum Bill Committee, said: "Almost two years on from the huge landslide support that the people of Scotland gave us, this poll also shows backing for the SNP has actually increased even further on those extraordinary levels."