The No camp are now six points ahead of the Yes campaign, down from 14 points in mid-August and 22 points early last month, excluding undecided voters.
The latest YouGov poll found that, excluding "don't knows", 53% of those questioned planned to vote No, while 47% would back Yes.
This compares to 57% for No and 43% for Yes in mid August and 61% for No and 39% for Yes at the beginning of last month.
Blair Jenkins, Chief Executive of Yes Scotland, said: "This breakthrough poll shows that Yes has the big momentum - it's an all-time high for Yes support in a YouGov survey so far, and an eight-point swing from No to Yes in just three weeks. We only need another three-point swing to achieve a Yes for Scotland on September 18.
"Scotland is one of the wealthiest countries in the world. More and more people are beginning to realise that a Yes vote is Scotland's one opportunity to make that enormous wealth work better for everybody who lives and works here, create more jobs here, and protect vital services such as the NHS from the damaging effects of Westminster privatisation.
"While the No campaign press the panic button and blame each other for a series of blunders, Yes will get on with the job of persuading more of our fellow citizens - both No and undecided voters - that we need a Yes vote to put Scotland's future in Scotland's hands."
The latest poll for The Sun and The Times questioned 1,063 respondents between August 28 and September 1.
Better Together Campaign Director Blair McDougall said: "We need the silent majority who back a No vote to do their bit. Whether it's voting on the day, knocking on doors, making phone calls or speaking to friends and family, the silent majority should feel confident in speaking up. We will not be complacent for one second and will do everything we can to secure the brightest future for Scotland within the UK.
"The nationalists talk as if they are winning but the truth is this is yet another poll showing the campaign for Scotland to stay in the UK in the lead. We speak for the majority of Scots, but this poll confirms that if people want Scotland to stay in the UK then they need to vote for it.
"The choice facing Scots couldn't be starker. A No vote is a vote for the best of both worlds - more powers for Scotland, backed up by the strength of the larger UK economy. A vote for separation would be a leap into the unknown, putting our currency, jobs, pensions and NHS at risk. By saying No Thanks we can avoid the £6 billion of extra cuts to services like our schools and hospitals that the experts say would be needed with independence."
The YouGov findings mirror a Survation poll for the Daily Mail last week, the first major opinion poll since the second televised debate.
Excluding those who are undecided it found that support for No was at 53%, with Yes on 47%.