The survey, carried out days after Scotland voted "No", found 3% wanted to Wales to be independent.
But there was support for the idea of more powers being devolved to the Welsh Assembly, with 49% in favour.
ICM Research interviewed a random sample of 1,006 people in Wales over the telephone.
The survey found 12% wanted to see the assembly abolished.
The poll also found growing support for UKIP in Wales ahead of next year's general election.
Nigel Farage's party is on 14%, up seven points from the last BBC Wales poll in March.
Labour are down four points to 38%, with the Conservatives on 23% (down one), Plaid Cymru on 13% (down one) and the Liberal Democrats on 7% (down two).
Prof Roger Scully of the Wales Governance Centre at Cardiff University said: "Support for independence is the lowest I've seen anywhere.
"There has been a clear move towards supporting more powers, and some of the people who may have said 'independence' have gone in that direction.
"We're getting close to a majority saying they want things to go further.  There are also pretty low levels of support for abolition of the assembly - the extreme positions are losing out."
An analysis of the voting intention figures by Prof Scully suggests that, on a uniform swing, Labour would gain two seats in the capital - Cardiff North and Cardiff Central - at next May's general election.
The Conservatives would lose Cardiff North but gain Brecon and Radnor; Plaid Cymru would retain their three seats but the Liberal Democrats would be reduced to a solitary Welsh seat at Westminster - Ceredigion.
UKIP's level of support is too low to win their own seat, but high enough to have an impact on some marginal constituencies.
"UKIP's support is clearly going up," said Prof Scully.  "They are starting to reach the sort of level where they could make a serious difference in marginal seats."
He said most evidence suggests they are taking more support from the Conservatives than anywhere else, which could be of importance in Tory-held marginal seats like the Vale of Glamorgan and Aberconwy.
ICM Research interviewed a random sample of 1,006 Welsh adults aged 18 or over by telephone on 19-22nd September 2014. Interviews were conducted across Wales and the results have been weighted to the profile of all adults. ICM is a member of the British Polling Council and abides by its rules.
There is more on this story on Week In Week Out on BBC One Wales at 22:35 on Wednesday 24 September.
Support for Welsh independence has fallen to its lowest recorded level in the wake of the Scottish referendum, according to a poll for BBC Wales.