Update 3.15pm The Taoiseach has said work has already begun to make sure the border with Northern Ireland stays open.
Enda Kenny said it is the goal of the governments in Dublin, Belfast and Westminster to make sure a hard border does not return.
The European Commission said today that the future of the border would have to be part of discussions between the UK and the other 27 member states.
But in the Dáil the Taoiseach said the border was a key area on which the UK and Ireland shared a position: “We will continue to work urgently and intensively, to ensure that collectively we can ensure that the last two decades, are fully protected in whatever post-exit arrangements are eventually negotiated.
“All three administrations, share the common objective, of wanting to preserve the common travel area, and an open border on the island of Ireland.”
Update 2pm: The Minister for Public Expenditure has said the Government still intends to deliver planned tax cuts and spending increases for Budget 2017, despite the result of the Brexit vote.
Last week, the Ministers for Finance and Public expenditure revealed that Ireland has about €1bn which can be allocated to increased spending and tax cuts for next year's Budget.
However, uncertainty over how Britain's exit from the EU will be orchestrated has resulted in dips in the stock markets and the value of Sterling.
Minister Paschal Donohoe has said we need to use the coming weeks to plan how we will deal with Brexit: “The result in the United Kingdom is fundamentally changing the environment in which Ireland will be operating.
“That will have consequences for us here, and we now need to use the period of time that is ahead of us, while the United Kingdom are negotiating their withdrawal from the European Union, to plan for this new order and environment.
“And to make sure that Ireland is as secure as possible in a very changed environment.”
Update 1.40pm: The Minister for Justice has said the Brexit vote is very disappointing.
However Frances Fitzgerald said we now have to get on with the reality of the situation: “Well it’s going to be a whole new set of relationships north and south, east, west.
“I’m very disappointed personally, I am a huge believer in the European ideal, I think it has served us very well, I think it has served Ireland very well.
“We are very close to the UK and to see them taking this decision, I regard as very disappointing, but it is the reality.
“It is now getting on with the new set of relationships.”
Update 1.10pm: The Minister for Foreign affairs has moved to reassure anyone wishing to apply for an Irish Passport that the entitlements have not changed as a result of the Brexit vote.
Charlie Flanagan has said UK passport holders will continue to enjoy EU rights for the foreseeable future until a formal exit of the of the country is negotiated.
Minister Flanagan has said there is been a spike in interest in Irish passports as a result of the referendum - however he said there has been some exaggeration of the actual demand.
Over the weekend it was reported that Belfast's main post office ran out of Irish Passport application forms.
Earlier: The Taoiseach is putting Ireland on a collision course with some of the larger EU member states, saying there will not be a speedy exit by the UK.
Enda Kenny has said Ireland is ready for the challenges ahead from what he called the "political earthquake" of the UK vote to leave the European Union.
He has told the Dáil that he wants a common agreed response across parties, but warned it is not possible to fully implement contingency plans as we still don't know the proposed arrangements for Brexit or the exact timetable.
And the Taoiseach took a swipe at a meeting of the founding members of he EEC over the weekend where they pushed for a speedy exit by the UK: “I also want to make it clear that it is the European Council, the leaders of the different countries, under the direction of President Donald Tusk
“And not any other EU institution or sub-group which will have overall political control of the process involved here, this is important Ceann Comhairle.”
Meanwhile the Fianna Fáil leader Micheál Martin told the Dáil that all European Union states need to address the rise of "the far right": “Let no one be in any doubt about how this result came about.
“There are many who are trying to spin it and play down what they can see for themselves.
“We have heard this already from some groups in this house as they tried to claim the result for their own cause.
“But please stop the attempt to cover up what everyone can see: This is the result of a relentless campaign of a tax on Europe and the promotion of an anti-foreigner agenda.”