LONDON, May 12 (Reuters) - Political leaders locked in a bitter fight over Scottish independence have urged all Scots to reunite whatever the outcome of a referendum in September as the national church voiced concerns over rifts in society.

Both sides in the campaign have called for an end to rising levels of abuse and personal attacks and stressed the need to work together after the vote over the 307-year tie to England.

As the race tightens ahead of the Sept. 18 vote, the Church of Scotland said there was a danger of deepening division over whether to leave the United Kingdom and invited leaders of both sides to a post-ballot reconciliation service.

Opposition Labour's foreign spokesman Douglas Alexander, in a speech in Edinburgh on Monday, stressed the importance of ensuring Scotland does not divide more deeply after the referendum and a debate marked by "rancour and bad tempers".

"The result of the referendum on Sept. 18 will leave Scotland divided, with a significant minority of the population feeling disappointed," said Alexander.

"The obligation and the challenge will be to ensure that Scotland - whatever the outcome - comes together and does not divide more deeply in the aftermath of this historic choice."

His comments came after the incoming moderator or chairman of the Church of Scotland, Rev John Chalmers, unveiled plans for a service at St. Giles Cathedral in Edinburgh on Sept. 21 to talk about healing and unity to overcome hurt and division.

CALL FOR RECONCILIATION

"In the coming months, there is a danger the referendum will set people against each other, in their own community, their own street - even their own family," he said in a statement.

"It will be important for each side of this campaign to be magnanimous whatever the outcome and the Church of Scotland, as a national church, is well placed to bring people back together in a spirit of reconciliation."

Last week the major funders of the "Yes" to independence movement, Scottish couple Chris and Colin Weir, who have donated 3 million pounds of a huge lottery win of 161 million pounds to the secession campaign, made a public plea to end smears and personal attacks on both sides.

The debate over independence has become increasingly vitriolic as support for secession has nudged higher this year, although the nationalists still have ground to catch up.

The latest opinion poll by Progressive Partnership on Sunday found 34 percent support for indepence with 54 percent against and 12 percent undecided. A similar poll last November showed 27 percent would say "yes" to independence and 56 percent "no".

Alex Salmond, leader of the Scottish National Party (SNP) that has dominated the Scottish Parliament since 2011, has appealed to pro-union supporters to get involved if Scotland does vote for independence.

He has pledged to set up a cross-party team to negotiate terms with the UK government if there is a Yes vote and ahead of Scotland officially becoming independent in March 2016.

"More clearly than anything else, this demonstrates the wish of those of us on the 'yes' side to move forward in a consensual way once the people have spoken," he told supporters on Sunday.