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.

Last year, National Grid said it would use underground cables instead of pylons between Anglesey and the mainland in Gwynedd.
It said this would protect the Menai Strait's natural beauty.
The tunnel will be based on a network of power tunnels in central London.
Plans to connect Wylfa Newydd at Cemaes Bay to a substation at Pentir, Gwynedd, have been controversial, with campaigners concerned about the impact of pylons.
Aled Rowlands, external affairs manager for National Grid, said: "We are going to be using the same type of technology [that we are using under the River Thames] to go underneath the Menai.
"When we've listened to local people and to those specialist bodies, they've asked us to preserve the natural beauty that's already there.
"This option is more expensive than just putting cables across the top, but we think it's worth it.
"We are looking at probably over £100m just for this cable tunnel to take the link from Anglesey over to the mainland and into the National Grid.
"Here in London, we are digging through London clay. There are very different conditions around the Menai, with some rock and some limestone and some other things, so it's going to be a real engineering challenge for us.
"But at National Grid we do have an expertise in that.
"This project for 34km [in London] has taken about eight years. We think it will be... four or five to build those 4km from Anglesey across to Gwynedd."
Plaid Cymru MP for Arfon Hywel Williams and AM Sian Gwenllian have met with National Grid bosses to push for undergrounding the remaining 1km of electricity connection, for which pylons are currently proposed.
The National Grid has published an interactive map showing the proposed routes of the power cables.
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.
[NEXT_CONCEPT]
A tunnel under the Menai Strait, which will contain cables linking Wylfa Newydd nuclear power station to the National Grid, is set to cost £100m and take up to five years to complete.