Irish Prime Minister Leo Varadkar has ruled out formal three-way talks between the UK, Ireland and the EU regarding Brexit issues.
Theresa May told parliament on Monday that the UK and Irish governments and the European Commission will be "working together" to avoid a hard border between Northern Ireland and the Republic.
Echoing last Friday's major speech on Brexit, Mrs May said: "We will not go back to a hard border between Northern Ireland and Ireland.
"Neither will we break up the UK's own common market with a border down the Irish Sea.
"As Prime Minister I am not going to let our departure from the EU do anything to set back the historic progress made in Northern Ireland, nor will I allow anything that would damage the integrity of our precious union.
"The UK and Irish governments and the European Commission will be working together to ensure that we fulfill these commitments."
But before Mrs May's statement, Mr Varadkar said it was not in Ireland's interests to take part in formal talks and that what was needed was for Downing Street to produce more detailed proposals.
He said: "There won't be tripartite or three-way talks.
"What will happen is that there will be talks between the EU 27 and the UK, and Ireland is part of the EU 27 and we're much stronger by the way as one of 27."
"We will of course have negotiations about what could be done to avoid a hard border, but what we won't be getting into is a negotiation with the UK, or a three-way negotiation," Mr Varadkar added.
"That's not in our interest and not the way that this can be concluded."
The Taoiseach told RTE's Morning Ireland programme that he gave Mrs May's Brexit speech on Friday a guarded welcome, but that detail was now needed from the UK Government.
"What we want is not so much principles and aspirations and red lines," Mr Varadkar said.
"What we want is detail, written down in black and white that can be codified into law and that is what is required."
Amid the ongoing row over the border, Sinn Fein leaders Mary Lou McDonald and Michelle O'Neill travelled to Brussels on Monday to meet EU chief negotiator Michel Barnier.
After what she described as a "frank, productive and positive" meeting with Mr Barnier, Mrs McDonald voiced support for the Taoiseach's stance.
"The Taoiseach is absolutely correct to say that the substantive negotiation is between the British government on one hand and the EU member states collectively on the other and in our view it is important to maintain that dynamic," she said.
The Sinn Fein president added: "The ball is now in Mrs May's court, the ball is now in the court of the British, who say they don't like what they have seen, they don't like the solution advanced by Europe, if they don't like that we would like to know what is their solution."
Mrs May said that she wants a deal that "provides an enduring solution", "protects people's jobs and security" and "strengthens our union of nations and people".
But she said the UK and EU should also accept "hard facts" including that the UK is leaving the single market meaning "in certain ways, our access to each other's markets will be less than it is now".
She reiterated the need to "strike a new balance" adding: "We cannot escape the complexity of the task ahead.
"We must build a new and lasting relationship while preparing for every scenario.
"But with pragmatism, calm and patient discussion, I am confident we can set an example to the world."
"Yes, there will be ups and downs over the months ahead, but we will not be buffeted by the demands to talk tough or threaten or walk out - and we will not give in to the counsels of despair that this simply cannot be done, for this is in both the UK's and EU's interests," she said.
The PM added: "My message to our friends in Europe is clear: you asked us to set out what we want in more detail, we have done that; we have shown we understand your principles; we have a shared interest in getting this right, so let's get on with it."
Replying to Mrs May's statement, Labour leader Jeremy Corbyn said there have been "20 wasted months" since the EU referendum in which the "arrogance" of some of the Cabinet who said it would be the "easiest deal in history" has turned into "debilitating infighting".
He said: "We've seen set piece speech after set piece speech and yet the Prime Minister still cannot bring clarity to these negotiations and still cannot bring certainty to British businesses or workers.
"The Prime Minister's speech on Friday promised to unite the nation, but it barely papered over the cracks in her own party."