By Jenny Gross and Nicholas Winning 

EDINBURGH, Scotland--U.K. Prime Minister David Cameron on Wednesday made a late push in Scotland for a united Britain, appealing to voters' emotions as he sought to convince them not to choose independence in a referendum next week.

Faced with fresh polls suggesting the Sept. 18 vote will be much closer than long thought, Mr. Cameron skipped his normal schedule and dashed to Edinburgh to plead with voters to stick with the U.K. and keep the 300-year-old union intact.

Mr. Cameron said he would be heartbroken if the U.K. was torn apart. "In the last few days of the campaign, let's make sure the emotion and passion comes through," he told several hundred employees of Scottish Widows Investment Partnership in Edinburgh.

"I care hugely about this extraordinary country, this United Kingdom that we built together," he said.

Scrambling to head off the risk of Scotland breaking away, U.K. political leaders have offered Edinburgh an array of new powers. In a rare united front, the leaders of the other two main political parties also cleared their schedules to campaign in Scotland, though they were doing so separately.

Mr. Cameron tried to assure voters that if they vote against independence the Scottish government would receive additional authority over taxation and welfare, saying all three parties have a strong commitment to increased devolution from lawmakers in London.

"Voting 'no' is not voting for no change," he said. "It really would be the best of the both worlds."

Until now, Mr. Cameron has stayed at the fringes of the independence debate in Scotland, where he and his Conservative Party are widely unpopular. Some Scots say their desire for independence stems from distrust of Mr. Cameron and his party, which they view as elitist, disconnected from more left-leaning Scots and unsupportive of policies popular in Scotland, such as increased subsidies for the poor and disabled.

Sadia Ghani, a 25-year-old trainee actuary at Scottish Widows, said after Mr. Cameron's speech that she's still undecided about whether to vote 'yes' to independence.

"There's nothing he said that made me come away and say 'no,'" she said.

Polls released over the weekend and this week showed a surprise surge in support for independence, putting the two camps neck-and-neck. For months, the pro-union camp was strongly in the lead.

Mr. Cameron and other leaders in London been accused of not taking the possibility that Scots would vote for independence seriously.

Tony Loy, a 22-year-old carpenter in Edinburgh, said he thinks Mr. Cameron's campaigning in Scotland will be a waste of a trip because of his unpopularity."I think this will up the vote for "yes," said Mr. Loy, who is voting for independence.

The referendum is the culmination of years of desire for self-governance. Scottish secession would have a profound effect on U.K. politics: It would be a blow for both Mr. Cameron and his Conservative Party, as well as the opposition Labour Party, which relies on Scottish votes as a significant part of its base.

While the party leaders were campaigning in Scotland, lawmakers in London representing constituencies in England, Wales, Northern Ireland and Scotland also urged Scots to reject independence.

Speaking for the government in Mr. Cameron's absence, former Foreign Secretary William Hague, said Scottish parliamentarians had made an immense contribution for generations and Parliament couldn't imagine life without the Scots.

"All of us in the United Kingdom would be diminished by the breakup of the United Kingdom. We are something greater than the sum of our parts," he said.

Nationalists need a simple majority to end their union. The leader of the Scottish Nationalist Party, Alex Salmond, says Scotland would be officially independent by March 2016 if Scots vote "yes." Other say it would take much longer because of uncertainties surrounding independence, such as what currency Scotland would use and whether it could rejoin the European Union as an independent country.

Mark Bishop, a retired 64-year-old living in Edinburgh, said he was pleased Mr. Cameron was finally making an effort to reach out to Scottish voters.

"It is about bloody time," Mr. Bishop said.

Write to Jenny Gross at jenny.gross@wsj.com and Jason Douglas at jason.douglas@wsj.com

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