David Cameron has announced a drive to urge young people to register to vote in a clear sign of concern on the Remain side about a tight result in the referendum on Britain's membership of the EU.
Casting the referendum as "a vote for your life", the Prime Minister said young people still had until June 7 to sign up for a vote in the June 23 poll.
And he signalled he is ready to accept the slenderest of majorities as a victory for Remain, dismissing the argument that a close result could provide the platform for a swift re-run poll.
Ukip leader Nigel Farage has argued that anything short of a two-thirds majority for Remain would represent "unfinished business" and could provoke demands for an early second referendum from Leave advocates who feel the campaign has not been run fairly.
But Mr Cameron said it was "obvious" that a simple majority was enough to settle the question in a referendum and Brexit supporters had to stick to the rules.
Speaking during a visit to Japan for the G7 summit, Mr Cameron said: "There are very clear rules for referendums - you've got to get out there and win a majority."
Mr Cameron made no effort to conceal his anxiety that younger voters - who polls suggest are strongly in favour of continued EU membership - may not use their ballots on June 23.
Some pollsters believe a low turnout among the young is the most likely factor to produce victory for Brexit, which is more popular among older age groups.
"One thing on the campaigning front that is probably my greatest concern is doing everything we can in the next week in order to get people to register to vote, particularly young people," said the Prime Minister.
"This is absolutely a vote about their future. This vote will determine the sort of country, the sort of economy they grow up in, the sort of opportunities that they have.
"Definitely something that is concerning me is that the last election turnout among young people was more in the 40 percents rather than the 60 percents.
"This is a vote for their lifetimes, so we are going to do everything we can to encourage people to register.
"If you go to gov.uk, it only takes three minutes. We are doing everything we can to encourage people because it is a vote for your life."
Amid growing signs of unease among Tory ranks over the Prime Minister's handling of the campaign, Mr Farage said earlier this month that a narrow outcome could create an unstoppable demand for a second referendum.
"In a 52-48 referendum this would be unfinished business by a long way," said the Ukip leader. "If the Remain campaign win two-thirds to one-third that ends it."
But Mr Cameron insisted: "Obviously a referendum is based on a simple majority."
He added: "There's some people for whom their life's work and their core belief is that Britain must get out of Europe. I have respect for people who hold a very strong view and do everything in politics they can to pursue that.
"That's not obviously my view. I disagree with them, but I'm sure there will always be some people who will go on making that argument.
"There's very clear rules for referendums - you've got to get out there and win a majority."
Mr Cameron declined to predict the result of the crucial ballot, saying only: "I want to get as many votes as I can for the case of staying in a reformed European Union, but I'm not going to make any forecasts, any predictions. I'm just going to do everything I can over the next 30 days to convince people that voting to stay in is the right thing to do."
He insisted the Remain campaign was "making good progress" in the referendum battle, particularly with its economic argument that Britain was better off if it stayed in.
Suggestions from some Vote Leave supporters that they would be prepared to see the UK leave the European Single Market had helped persuade voters to switch to Remain, he indicated.
"I think one of the key moments of this campaign was when the Leave campaign said they wanted to leave the Single Market," said Mr Cameron.
"I think that has really crystallised the argument about the economy and the risks to our economy, to jobs, to growth and to prices if we leave."
He said Tuesday's report from the independent Institute for Fiscal Studies think-tank, which warned that Brexit could trigger two more years of austerity, offered "very, very powerful" support for the Remain case.