The TNS snapshot placed support for a Yes vote at 32 per cent, up three per cent on a similar survey in January, while support for a No vote was 46 per cent, up four. Don't knows were on 22 per cent, down seven. When the undecideds are removed, the split is 41 per cent for independence and 59 per cent against.
The average of the polls in June put Yes on 37 points, No on 48 and don't knows on 15. With just over two months to go to Referendum Day, this means the Yes camp, which saw the polls narrow earlier this year, has still to make the breakthrough it needs to win on September 18.
Mr Darling, who leads the pro-UK Better Together campaign, said: "We're making progress, which is why, with increasing confidence, I can say we will win, provided we continue to get our arguments across."
Blair McDougall, Better Together campaign director, was equally upbeat, saying: "It's clear the closer we get to the referendum, people are starting to think seriously about the consequences of separation for our pound, pensions and public services … We are very confident that on the basis of the facts Scots will say No Thanks to separation."
But Blair Jenkins, chief executive of Yes Scotland, claimed the poll, commissioned by leading Scottish businessman and philanthropist Sir Tom Hunter, was very encouraging for the Yes campaign.
He pointed to how "headline support" was more than 40 per cent, noting how support for independence was up six points while backing for the status quo was down below 25 per cent.
"More and more people want to see a Scottish Parliament with full powers and undecided voters increasingly see that a vote for independence is the only way to secure that," insisted Mr Jenkins.
The figures he referred to relate to three options those polled were offered. Support for independence was up from 24 per cent to 30; support for devo-max rose from 35 per cent to 39; and backing for the status quo was down from 31 per cent to 22.
While the pro-UK parties insist a No vote does not represent the status quo but more devolved powers, it was suggested to Mr Darling there was no certainty about this.
The former Labour Chancellor stressed how the three main UK parties had delivered in the past and added: "The three parties have said that this is what they will do and it's terribly important that, having said it, they stick to it."
The TNS poll also found:
l 45 per cent of people do not trust either the UK or the Scottish Government on the economy. 18 per cent said they trusted the former and 25 per cent the latter;
l 63 per cent said they had given up listening to the referendum debate;
l 73 per cent said it was hard to know who to believe;
l 68 per cent said that they were unaware more powers were already due to come to Holyrood in 2016.
Sir Tom commissioned the poll for his impartial Scotland September 18 foundation aimed at providing open debate and information.
In response to it he said: "The deluge of contradictory 'evidence', lack of knowledge combined with distrust in both governments' forecasts, leaves many at best confused and at worst sick and tired of the debate.
"This is an appalling state of affairs brought about by both sides of the debate and is an affront to our democracy." He added: "I would call on both sides to step up their games, be honest with their predictions and engage the population over the priorities that matter to them most in voting Yes or No: healthcare, employment and the economy, according to this poll.
"The future of our nation is rightly in the hands of our population, but that population is being enveloped by a thick fog of obfuscation."