BELFAST (Reuters) - The British and Irish prime ministers held talks with Northern Ireland’s feuding political parties on Monday and said later they were hopeful the province’s year-old political stalemate would soon end.

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The British province has been without a devolved executive for over a year since Irish nationalists Sinn Fein withdrew from a power-sharing government with their arch-rivals, the Democratic Unionist Party (DUP).

Failure to reach a deal would be likely to lead to the introduction of direct rule of the region from London for the first time in a decade and a diplomatic dispute over what role the Irish government should then have in the region.

It would further destabilize a delicate balance between Irish nationalists and unionists who, until last year, had run the province since 2007 under the terms of the 1998 Good Friday peace accord which ended three decades of violence.

“The differences that exist between the DUP and Sinn Fein are not insurmountable, and we are very hopeful that those two parties will be able to come to an agreement this week,” Irish Prime Minister Leo Varadkar told journalists after the talks.

Britain’s Theresa May called on the parties to make one final push. “It should be possible to see an executive up and running in Northern Ireland very soon,” she added.

The DUP and Sinn Fein have failed to reach agreement on a number of issues, including the introduction of same-sex marriage, which is illegal in Northern Ireland but legal in the rest of Britain and Ireland.

Rights for Irish-language speakers and funding for inquests into deaths during the decades of Protestant-Catholic sectarian violence before 1998 have also proved contentious.

BREXIT AND THE BORDER

“We don’t believe there is anything insurmountable left to resolve,” Sinn Fein leader Mary Lou McDonald said.

DUP leader Arlene Foster said very good progress had been made, but that agreement would have to “recognize all cultures” and not “place one above the others,” an apparent reference to Irish-language rights.

The meetings come as Britain bids to secure a breakthrough in talks with the European Union on the terms of its exit from the bloc, which could have dramatic implications for Northern Ireland.

The future trading relationship will help determine whether physical infrastructure will be required to control the border between the region and the Republic of Ireland, which will become an EU frontier.

EU negotiator Michel Barnier last week warned that if Britain proceeded with plans to leave the EU’s single market and customs union, border checks would be unavoidable.

Varadkar on Sunday called for “clarity and urgency” from London in laying out exactly what kind of a post-Brexit deal it wants, but on Monday struck a more positive tone saying that “things are going in the right direction.”

May, who is hoping to seal a transition deal with the EU next month to smooth Britain’s exit, on Sunday said she would set out what it wants from Brexit in a series of speeches over the next few weeks.