He added: "I have only a limited recollection of the border and the Troubles, but I know that I, like almost everyone else on this island, do not want to go back to that", he said.
Ireland's Foreign Minister and former MEP Simon Coveney, whom Mr Varadkar beat in their party's leadership race, said the governments should not be talking about technological solutions for manning the Border but political ones and "flexible and imaginative" ones at that.
He said it was vital the Irish Government engaged with the UK Government on an ongoing basis.
The good news is that there was a breakthrough in this year's Assembly elections when Pat Catney of our sister party, the SDLP, gained a seat from the DUP in Lagan Valley and broke their stranglehold on the legislature with a clear commitment to support equal marriage.
This was earlier confirmed by Mr Varadkar when, during a question and answer session at Queen's University, he said: "It's not a proposal that I'm tabling, and I wouldn't like us to be in that position".
Mr Varadkar will also stress the importance of hearing the voice of Northern Ireland's elected representatives on Brexit and encourage parties to work to restore the Executive.
Brexit is the challenge of our generation.
In October, the Irish premier will, as part of the EU Council, discuss whether sufficient progress has been made in the Brexit negotiations for them to move onto future trade.
"Every single aspect of life in Northern Ireland could be affected by the outcome, jobs and the economy, the border, citizens" rights, cross border workers, travel, trade, agriculture, energy, fisheries, aviation, European Union funding, tourism, public services, the list goes on'.
For some, he is the Irish version of French president Emmanuel Macron: a dynamic young leader committed to Europe; for others, he is a unsafe right-wing ideologue "who threatens Ireland's centrist consensus", as Vincent Boland put it in the Financial Times.
'Who will speak for Northern Ireland and her 1.8 million people?
Divorce talks between Britain and the bloc are underway, but European Union officials say progress must be made on resolving issues around the Irish border, the size of Britain's exit bill and the status of European Union nationals in the United Kingdom before the two sides can start hammering out their future economic relationship.
Ireland Prime Minister Leo Varadkar said that he did not want to see an economic border between his country and any part of a post-Brexit Britain.