Just over half of people questioned said they would still support leaving the European Union even if they knew it could lead to Scotland becoming independent.
The poll for the Daily Telegraph found that while 59% of people across the UK were opposed to Scotland leaving the UK, almost the same number said Brexit was more important than the potential break-up of the Union.
That was the stance of 59.9 per cent of people surveyed by ORB. Just over a quarter (26.6 per cent) disagreed with this, while 13.5 per cent said they did not know.
The survey of 2,000 voters was carried out on Wednesday and Thursday after Scotland's First Minister Nicola Sturgeon announced plans to stage a second independence referendum to give voters a choice between leaving the UK and a "hard Tory Brexit".
Prime Minister Theresa May has dismissed the call, saying now is not the time for a fresh ballot.
Ms Sturgeon says Scots should be allowed to vote on staying in the UK some time between autumn 2018 and spring 2019 - arguing this would allow them to make an "informed choice" when the terms of the Brexit deal are known.
But 58.5 per cent agreed that having a second Scottish independence referendum before Brexit would undermine the UK's ability to negotiate with the EU, according to the poll.
Meanwhile 50.5 per cent said they would still back Brexit even if they knew it could lead to the end of the United Kingdom, with 37.8 per cent saying they would not support an exit, and 11.7 per cent did not know.
While a majority of people across the UK voted to leave the EU in 2016, almost two-thirds (62 per cent) of Scots backed Remain.
SNP ministers insist this gives them a "cast-iron mandate" for a second independence referendum, after the party's manifesto for the 2016 Holyrood elections said another poll could be held if there was a "material change in circumstances" since the 2014 vote, such as Scotland being removed from the EU against its wishes
Additional reporting by Press Association.