The survey found nearly one in three voters (31%) is a "don't know", less than a year from polling day.
The figure is up three percentage points compared with TNS BMRB's previous poll in August and is the highest recorded in any recent survey.
The two competing campaigns last night agreed the referendum was wide open as they promised to redouble efforts to persuade the key group of undecided voters.
They also said they would provide more information, in response to another finding showing 44% of voters felt poorly informed about the debate.
The poll of more than 1000 adults, conducted in late September and early this month, revealed a strong lead for the pro-UK campaign in line with most recent tests of public opinion.
One in four (25%) backed independence, the same level seen in August, compared with 44% who were opposed.
The rise in the number of don't-knows was matched by a fall of three percentage points in support for a No vote.
Of those certain to vote - often regarded as a more accurate pointer - 28% planned to vote Yes, 50% No, and 22% were undecided.
In a separate poll of people in England and Wales by TNS BMRB, results showed 53% of people wanted Scotland to remain part of the UK, compared with 22% who felt it should become independent. A further 25% were unsure.
Independence would have little impact on the Scottish tourism industry, the poll also suggested, with 77% of English and Welsh respondents saying it would make no difference if they were planning a holiday north of the Border.
Tom Costley, head of TNS in Scotland, said the focus on the referendum as it passed its year-to-go milestone last month had failed to shift public opinion significantly. He added: "The lack of information is particularly evident among the 'don't know' group, of whom 60% indicated they were very much lacking the information they need.
"While we recognise that not all those in the 'don't know' group are engaged in the debate, just under half say they are certain to vote.
"Both camps need to be more convincing in their communications to this important constituency in order to make progress."
TNS BMRB has tended to find higher levels of don't-knows than other polling organisations.
In its fieldwork the company asks voters "how they intend to vote" next year, a formulation polling expert Professor John Curtice suggested may make it easier for people to say they were undecided.
Comparing the various polls this year, the number of don't-knows has ranged from 9% to 31%.
A spokesman for the pro-independence Yes Scotland campaign said: "The most significant figure in the TNS poll is the increase in the number of undecideds.
"We know from our own research that the more people learn about independence the more likely they are to vote Yes because they realise that having a government with full powers they vote for in Scotland is the best way to ensure our undoubted wealth is used to build a fairer, more prosperous country.
"We know we have a lot of hard work to do over the next 11 months to get our message across to even more people, and we are confident that on September 18 next year Yes will win."
A spokesman for the pro-UK Better Together campaign said: "While it is always better to be ahead in the polls than behind, it is clear that it is still all to play for in the referendum.
"We have to make sure that we fight for every vote between now and polling day. If people believe it is better for our jobs, our pensions and our taxes to stay part of the United Kingdom, then they have to come out and vote for it."