Taoiseach Leo Varadkar has presented US President Donald Trump with the shamrock bowl during a traditional ceremony in the White House.
In a ceremony at the White House, the Taoiseach said the Irish Government will continue to work with the Trump Administration to find a solution to the issue of the undocumented Irish and added that the government is ready to do a deal.
Mr Varadkar also said that "the United States has helped build modern Ireland, one that is prosperous and at peace, self-confident about our place in the world."
Mr Trump told reporters the Taoiseach is "doing a great job" as he greeted him outside the White House ahead of the bowl presentation this evening.
It comes after Mr Trump said he would like to visit Ireland soon saying he has a property there that he may never get to see again.
Mr Varadkar met Mr Trump as part of a traditional programme of events ahead of St Patrick's Day.
Mr Trump described the border with Northern Ireland as "interesting" and said that the two leaders would discuss trade, military and cyber issues.
Asked about the Irish border he said "we have two interesting borders and it will be interesting to see what happens there".
He said it was his honour to have the "very popular Prime Minister of Ireland" visit him.
He described the Irish living in the US as "tremendous, wonderful people that we love".
President Trump said the relationship between the US and Ireland was "outstanding" and only getting better.
The meeting between the two leaders lasted around 40 minutes.
Taoiseach Leo Varadkar has presented US President Donald Trump with the traditional bowl of shamrock pic.twitter.com/qRi8nFkbHi— RTÉ News (@rtenews) March 15, 2018
Afterwards, Mr Varadkar described it as a "good" meeting, and said a firm date for his visit to Ireland had not yet been set - but that the President had a "standing invitation".
He said during the discussions Mr Trump raised the issue of the Irish "undocumented" - those currently living in the US without legal permission to remain.
"It was something that was very much on his mind," he said.
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"We have a measure of support and degree of enthusiasm from the administration to work on a solution for thousands of Irish people who are here undocumented but who are hard-working, tax-paying people who are very loyal to America."
Mr Varadkar said Brexit's impact on the Irish border also featured in the talks, which also involved Vice President Mike Pence.
"The president was very aware of the issues that could affect Northern Ireland if there is a return to a hard border and I think will be very much on our side in working for a solution to make sure that doesn't happen," he said.
Trade was also discussed.
"The president has enormous concerns that the United States isn't being treated fairly when it comes to trade by China and by Europe," Mr Varadkar said.
"I put across the view that maybe the best way to resolve that is for a new deal, a new trade deal between America and Europe, and the president seemed very open to that."
Mr Varadkar was asked about Mr Pence's decision not to allow the media to attend their scheduled meeting at his residence on Friday morning.
The Taoiseach has pledged to raise issues around the Vice President's controversial stance on LGBT rights.
Mr Varadkar said he would have preferred if the cameras were allowed in to document their comments, but he added: "It allows us maybe to have a frank conversation that's easier to have without the media present."
He said he would highlight the fact that Ireland has a balanced economic relationship with the US, as investment and jobs flow in both directions, and that there should be no new barriers to trade.
After Britain leaves the EU, Ireland will be the only English-speaking country in the union and the country closest to America in terms of culture and business systems.
This, the Taoiseach said, can give Ireland an enhanced role as a bridge between the EU and the US.
After his meeting with Mr Trump, Mr Varadkar attended a lunch on Capitol Hill where he explained how he got a surprise call from Mr Trump a few years ago.