Theresa May promised no hard border during a trip to Dublin this week
She also pledged to keep the Common Travel Area with mainland Britain in place.
The Irish Taoiseach Enda Kenny said the pair enjoyed a “frank” discussion about Brexit, which is set to have a serious economic knock-on effect with the Republic.
Mrs May also paid tribute to the “special and unique relationship” between the two countries, aided by decades of cross-border migration and the UK’s status as the Republic’s biggest trading partner. Ireland is the UK’s seventh biggest partner.
In an editorial for today’s Irish Times, Mrs May said: “I am in Dublin this week with a very clear message: that the UK government is committed to preserving and enhancing the unique relationship between our countries and our peoples.
“Geography, history and the close family ties and bonds of affection that unite the UK and Ireland mean that there will always be a special and unique relationship between us.”
She acknowledged the economic, social and security “concerns” about Brexit in Ireland, especially regarding the contentious issue of a potential hard border with Northern Ireland, and pledged to address them fully.
Theresa May said she would not "turn back the clock" on Ireland
The PM said: “First of all, the vote to leave the EU was no rejection of the values we share with our European friends, least of all Ireland.
“We do not want to turn the clock back to the days when Europe was less peaceful, less secure and less able to trade freely.
“I am wholeheartedly aware of what is at stake in Ireland and Northern Ireland as the UK leaves the EU.
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“Second, I want to reassure people about the travel arrangements between our two countries – particularly when it comes to the Border.”
However, the statements issued by Mr Kenny after the meeting were nowhere near as optimistic – and came amid a background of anger after Mrs May snubbed an invitation to speak to the Irish Parliament.
Theresa May upset politicians in Ireland and the UK by refusing to speak to the Dail
Mr Kenny said: “I made clear that, in my view, any manifestation of a hard border would have very negative consequences.
“We had a frank but constructive discussion and identified a number of areas where we have further work to do together."
Enda Kenny warned Theresa May Ireland would not accept a hard border
Mrs May's decision to reject an invitation to speak to the Irish parliament, which would have seen her become just the second ever PM to do so, was roundly panned in Ireland and Britain.
Former Lib Dem leader Nick Clegg said it showed Mrs May had "lost the plot", due to Ireland's status as the UK's last real EU ally.