If a hard Brexit for most of the United Kingdom and a soft Brexit for Northern Ireland is out of the running because of the DUP, and a hard Brexit for the whole of the United Kingdom including Northern Ireland is out because of the border issue, that leaves only a soft Brexit for all of the UK.
If Mrs May has managed to square off the DUP and Ireland on the border issue, it appears likely EU leaders at December 14's European Council summit will approve "sufficient progress" has been made for talks to move on to phase two, on trade.
But Mr Grayling, a key Leave campaigner in the 2016 referendum, said: "We don't have to have, and we've never said we will, and we don't want, to have a situation where in future our laws are identical to the European Union".
The government says it is "optimistic" about an Irish border agreement as pressure mounts to find a Brexit deal.
Commission chief Jean-Claude Juncker's chief of staff Martin Selmayr tweeted a picture of white smoke - the sign used by the Vatican to signify the election of a new pope - shortly after May's arrival.
Hopes of a deal rose after Leo Varadkar, the Irish Prime Minister, told Ms May he was willing to look at a new text and potentially shift Dublin's position.
Rain and sleet Thursday, light snow possible overnight in Houston area
Pedestrians stroll past Jones Hall in the rain on Wednesday as a cold front barged into Houston, making it feel more wintry. While it will be a cold rain that falls through the day, temperatures will be too warm for any wet snow to mix in.
"In Northern Ireland we guarantee there will be no hard border", May told a press conference with Juncker. May is struggling to balance those demands against the concerns of Northern Ireland's Democratic Unionist Party, which she relies on to support her government in Parliament.
After talks in Dublin, Irish Taoiseach Leo Varadkar and Dutch PM Mark Rutte made clear the European Union would not compromise and allow the Irish border to be kicked down the road to phase two of the talks, even under threat of Britain crashing out with no deal or divorce negotiations dragging on to 2018.
On Thursday, EU Commission spokesman Margaritis Schinas had dismissed British newspaper reports that the Sunday deadline could be extended into next week as "not correct".
First Minister Nicola Sturgeon said "the real lesson" of the past week was that Scotland "will always be at the mercy of reckless decisions taken by Tory governments at Westminster" unless it becomes independent.
The government does not agree, and says it is confident negotiations can be completed by March 2019.
"I don't believe that the City will fall apart and that everyone will flee".
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