The survey suggested backing for a split rose in the wake of Theresa May confirming the UK will leave the single market, when it quits the bloc.
First Minister Nicola Sturgeon has signalled she is ready to launch a new bid to break up Britain in "the next few weeks" if Mrs May does not back her plan for a bespoke deal.
The BMG poll for the Herald found that 49 per cent of Scots support independence compared with 51 per cent who back the Union, when don't knows are removed.
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Most Scots do not want another independence referendum before Britain leaves the EU
When the 10 per cent undecided are included, 43 per cent would vote Yes and 45 per cent No.
We don't believe that there should be a second referendum
Theresa May spokesman
The gap has narrowed since a similar poll last month which showed 45.5 per cent were Yes supporters and 54.5 per cent backed No.
But the survey of 1,067 adults found 48 per cent against a rerun of the 2014 plebescite before Brexit compared to 38 per cent in favour.
Thirteen per cent were undecided – making 56 per cent against and 44 per cent for when the "don't knows" are removed.
Nicola Sturgeon has signalled she is ready to launch a new bid to break up Britain
It came amid reports the Prime Minister is drawing up a secret strategy to combat the next independence campaign in the beleif Ms Sturgeon will demand a referendum for August next year.
But Whitehall sources insisted there was no formal contigeny planning underway.
And asked about the report, Mrs May's spokesman said: "We don't believe that there should be a second referendum.
"There has been a referendum. It was clear, decisive and legal. Both sides agreed to abide by the result of that referendum."
Meanwhile, SNP Westminster leader Angus Robertson accused the Prime Minister of failing to keep her promise of securing "an agreed UK-wide approach" before starting the Brexit process.
‘We don't believe that there should be a second referendum’, says Mrs May's spokesman Nicola Sturgeon's top tweets Sun, July 24, 2016
With over 391 thousand followers on Twitter, First Minister of Scotland, Nicola Sturgeon is a prolific tweeter, here are her best quotes.
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Nicola Sturgeon, Leader of Scottish National Party and First Minister of Scotland
He said Mrs May should not "carry on regardless" without securing a UK-wide compromise.
Mrs May insisted she was committed to working with the devolved administrations adding: "I would also remind him of this point because he constantly refers to the interests of Scotland inside the European Union.
"An independent Scotland would not be in the European Union."
Scottish Labour MP Ian Murray said the SNP was "fixated" on independence, insiting that "people in Scotland are very clear that they don't want another referendum".
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