In her first major Brexitspeech since the wave of ministerial resignations which followed her Chequers deal, the Prime Minister described the white paper proposals as "a significant development of our position ... a coherent package".
In a swift response, the EU's chief Brexit negotiator Michel Barniersaid the bloc was open to alternative proposals on the border but insisted it be a legally watertight agreement as part of the withdrawal treaty.
The EU's plans to retain frictionless trade and travel between the island's two parts include keeping Northern Ireland closely aligned with the bloc's customs union, single market and Value-Added Tax system.
"This requires, in particular, a legally operative backstop, an all-weather insurance policy to address the issue of Ireland and Northern Ireland", Barnier said.
Barnier accepted that the White Paper had brought about "constructive discussions" and they are "not that far" from reaching an agreement.
The border issue is one of the most disputed parts of the Brexit negotiations.
Barnier, who met new UK Brexit minister Dominic Raab on Thursday, also reminded Britain there could be no progress on future relations until all elements of its exit were tied up - notably ensuring no hard border between Ireland and Northern Ireland.
The EU sees securing the new external border of the EU after Brexit as a key issue in the withdrawal agreement that needs to be negotiated by October to have enough time for ratification by the British and European parliaments before Brexit takes effect next March.
Theresa May's political future has been uncertain since her "white paper" policy document on post-Brexit relations with the European Union triggered resignations from the Cabinet.
"The protection of the peace process and upholding our binding commitments in the Belfast Agreement are grave responsibilities", she said.
"We want to de-dramatise this, talk it all down, and so there was surprise at the aggressive tone", one diplomat said.
It is very encouraging that the Prime Minister came to Northern Ireland to listen and to understand the concerns of business, on maintaining a common rulebook over goods and agri-foods and how the UK Government can not simply "wash its hands" of the matter.
The EU thinks a Brexit no-deal scenario would be a bureaucratic, practical and financial nightmare for Europe.
May also said: 'I am passionate about that brighter future and the possibilities that are within our grasp.
British Prime Minister Theresa May has endured swift condemnation from conservatives for agreeing to a "soft Brexit" deal that keeps the United Kingdom too closely aligned with the European Union.
The EU has proposed that Northern Ireland stay aligned with the bloc after Brexit if no other solution to the hard border can be found.
All outstanding problems - including the most controversial and intractable, the Irish border question - are supposed to be solved by October, when the European Union expects a final deal to be reached on what Brexit will look like, including the UK's ongoing relationship with Brussels.
She said thousands of people who cross and re-cross between the United Kingdom and Ireland in the normal course of their daily lives can not be subject to a hard border as they go to work, visit a neighbor, or go to the supermarket.
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