The Panelbase survey of 1,020 voters for the Sunday Times found 51 per cent are not supportive of a second referendum within the next year or two.
The poll, carried out last week, found support for another referendum before Brexit has fallen from 43 per cent following the vote last June to just 27 per cent.
A new poll shows that Scots do not wish to leave the UK in an independence referendum
While 61 per cent said they believe the UK should remain in the EU, 39 per cent want to leave, with support for Brexit at 35 per cent of those who previously voted Yes.
Scotland is now sending a very clear message to Nicola Sturgeon – we don't want your second referendum
Around a third (31 per cent) of voters oppose Scotland applying to join the EU in the event of independence, compared to 48 per cent who are supportive and 21 per cent who do not know.
Overall, the poll found support for independence is slightly up on the 2014 result at 46 per cent, with the No vote at 54 per cent.
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Nicola Sturgeon visits Glaxo Smith Kline.
51 per cent of people did not support a referendum in the next year
Mr Sturgeon's predecessor, Alex Salmond told Sky News that if the Prime Minister "bulldozes" her way to a deal which hit jobs and investment in Scotland, another plebiscite "becomes very likely".
He said: "In my view it should be held within the two-year period [for Brexit talks] so Scotland could have the option of continuous membership of the EU or European Economic Area."
But Scottish Tory leader Ruth Davidson said: "Scotland is now sending a very clear message to Nicola Sturgeon – we don't want your second referendum.
The SNP was blasted for not dealing with 'day-to-day' problems
"People desperately want the SNP to focus on the day job, not yet more division and uncertainty."
A spokeswoman for the SNP said: "These figures also show that people understand the economic damage that is threatened by the Tories' hard Brexit obsession, which risks tearing us out of a single market eight times bigger than the UK's alone."