The BMG survey, for The Herald newspaper, found 43 per cent of Scots support independence, with 45 per cent saying they oppose a breakaway from the union.
When ‘don’t knows’ are excluded, 49 per cent backed independence - similar to polls in January and February this year.
Last month First Minister Nicola Sturgeon was dealt a blow when Theresa May announced “now is not the time” for a second referendum on Scottish independence.
Sturgeon had predicted support for an independence Scotland would rise following the move by Westminster.
Her predecessor Alex Salmond suggested the refusal to agree to a referendum would “backfire spectacularly.”
Dr Michael Turner, head of polling at BMG Research said: “Out latest poll shows public support for an independent Scotland remains on a knife-edge.”
An SNP spokesman said the results showed support for independence remained “strong and steady”.
Scottish Greens MSP Ross Greer said:” This latest poll shows a dead heat between Yes and No and that’s before the potential re-election of an extremist Conservative Government and its devastating hard-Brexit agenda.”
But John Lamont, Scottish Conservative chief whip said: “This poll shows once against there has been no surge in support for independence.”