The DUP has said British Prime Minister Boris Johnson rejected the idea of a Northern Ireland-only backstop during a meeting this evening at Downing Street.
DUP leader Arlene Foster and parliamentary leader Nigel Dodds both met Mr Johnson.
Afterwards, Ms Foster released a statement saying Mr Johnson had again rejected the idea of a special economic zone for Northern Ireland.
"The Prime Minister rejected a Northern Ireland-only backstop in a letter to Donald Tusk on 19 August," a statement said.
"It is undemocratic and unconstitutional and would place a tariff border between Northern Ireland and the rest of the United Kingdom. That would be unacceptable.
"During today's meeting, the Prime Minister confirmed his rejection of the Northern Ireland-only backstop and his commitment to securing a deal which works for the entire United Kingdom, as well as our neighbours in the Republic of Ireland."
The idea of a Northern Ireland-only backstop had been reintroduced as the parliamentary impasse meant a no-deal Brexit became increasingly likely, and no agreed solution on how to manage the Irish border has yet been offered during negotiations between the EU and UK.
Taoiseach Leo Varadkar has not ruled out the idea and Fianna Fáil leader Micheal Martin said yesterday that he felt it was a viable solution.
The DUP has been against the idea since its inception, labelling it "anti-democratic and unconstitutional".
Ms Foster said a deal was possible without Northern Ireland being subject to different customs regulations to other parts of the UK.
"History teaches us that any deal relating to Northern Ireland which cannot command cross-community support is doomed to failure," she added.
"That is why the Northern Ireland backstop is flawed. Not one single unionist MLA in the Northern Ireland Assembly supports it.
"We want to see the referendum result implemented. Those blocking Brexit are causing uncertainty but more worrying they are damaging democracy by ignoring the United Kingdom's decision.
"A sensible deal between the United Kingdom and European Union, which respects the economic and constitutional integrity of the United Kingdom, is the best way forward for everyone."
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Earlier, a spokesman for Mr Johnson also said Britain was not seeking to negotiate a Northern Ireland-only backstop arrangement.
Officials in Brussels say that to get a Brexit deal done, Mr Johnson may need to go back to the plan, which involves applying different border rules to Northern Ireland from the rest of the country to help keep goods flowing across the Irish border.
"We are not seeking a Northern Ireland-only backstop," the spokesman said.
No-deal Brexit 'not in the interest of the EU or UK' - von der Leyen
The European Commission's incoming chief said a no-deal Brexit is not in the interest of the EU or the UK.
Ursula von der Leyen said Brexit "should it happen" would be only the start of a new relationship between the UK and the EU, and "not the end".
She said preparations for a no-deal were done and she wanted the EU's chief Brexit negotiator, Michel Barnier, to stay on in his role.
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"The Brexit, should it happen, is not the end of something but it's the beginning of our future relationship," she said as she named Phil Hogan to take charge of EU trade, including post-Brexit talks with Britain on a free trade deal.
Asked about Mr Johnson being pressured by his parliament to ask for another extension to Brexit - a third, to run until the end of January 2020 - Ms von der Leyen said that entirely up to the UK government.
"The next steps are completely in the hands and the decision of the United Kingdom. So I will not comment on their decisions and the next steps they might take," she said.
"So we'll see how things will proceed."