The Taoiseach (Irish prime minister) has announced that the Dáil (Irish parliament) will be recalled on Monday to discuss the impact of Brexit.
The result was not the outcome the Republic of Ireland wanted.
Enda Kenny said the Dáil would publish a summary of the key actions it will take to address the issues arising from the decision of the British electorate.
He said the UK vote to leave the European Union will require "careful consideration" by the Dáil.
"I am very sorry that the result of the referendum is for the UK to leave the European Union. However, the British people have spoken and we fully respect their decision," said Mr Kenny.
"I want to assure the Irish public that we have prepared to the greatest extent possible for this eventuality.
"There will be no immediate change to the free flow of people, goods and services between our islands."
With regard to the prospect of a border poll - called for by Deputy First Minister Martin McGuinness - Mr Kenny said: "We have more serious issues to deal with in the immediate term and that is where our focus is."
The Irish government, which stayed neutral in the Scottish independence campaign, actively encouraged Irish citizens in the UK to vote to remain in the EU.
"We have previously set out our main concerns in the event of Brexit," said Mr Kenny.
"These relate to the potential impacts for trade and the economy, for Northern Ireland, for the common travel area and for the European Union itself.
"In the medium term, a related concern is that of the Common Travel Area between Britain and Ireland.
"For our part, the Irish Government will do our utmost in upcoming discussions to maintain the Common Travel Area and minimise any possible disruptions to the flow of people, goods and services between these islands."
Micheál Martin, the leader of the main opposition party Fianna Fáil, has called on the Irish government to be centrally involved in the UK's divorce negotiations to minimise the repercussions and to make clear that despite the historic close links and trade relationships between the two states, Ireland will not be following the British example.
With uncertainty over what happens next and how long the process of UK EU withdrawal will take, there will be concern in Dublin about the implications of the vote for the border with Northern Ireland.
There is also the issue of the impact of a weakening sterling and the effect on trade which is estimated at more than 1bn euro (£819m) between the Republic of Ireland and Northern Ireland every week.
In the course of the campaign, Irish ministers disagreed with Northern Ireland Secretary Teresa Villiers about whether the border would be "hard" or "soft", if the British decided to leave,
Dublin said some form of border control - whether on the border or at airports and ports - was likely to be reintroduced.
The Irish capital city's chamber of commerce has called on the Irish government and Irish MEPs to "act decisively" with the EU to ensure that Irish business interests are protected.
Gina Quin from the chamber called the result "disappointing" and "a major challenge for Irish business, particularly given that the UK is one of Ireland's most important trading partners".
"The focus of everyone in Ireland must be on ensuring that the Irish voice is heard loud and clear in the negotiations which will now take place," she said.
"We must ensure that our trade with the UK and the movement of talent and tourists between our two islands remains on a stable footing."
Meanwhile, one of the towns that sits close to the Republic of Ireland side of the border and relies heavily on cross border trade is braced for the fallout of that UK decision.
In a statement, Dundalk Chamber of Commerce said it was "disappointed" by the the UK vote to leave the EU.
"We would stress that, in the short term, business will continue as before," said Paddy Malone from the chamber.
"The border with Newry will not change in the short term and we would hope that the Irish government will do all in their power to minimise disruption for manufacturers, transport, retail, tourist and employees who live and work in different jurisdictions," he said.
"We must strive to remain good friends and the EU needs to continue to support the peace process."