Scotland's First Minister Nicola Sturgeon told the Seanad that Brexit "creates a challenge that is not our challenge, or of our making".
"How we respond to June's vote will guide us for generations to come," she added.
The Scottish leader is the first foreign head of government to address the Seanad.
It is the culmination of a two-day visit to Ireland, aimed at strengthening economic links between the two countries.
She said that she visited Trinity College to see the Book of Kells last night and it is a reminder of the historic ties between the countries going back to St Colmcille visiting Iona.
Much of the modern history of both countries has been shaped by emigration from beyond these islands and both countries can take great pride in what Scotland and Ireland has achieved, she added.
The Scottish leader also said that Europe is facing its greatest refugee crisis since World War II.
Scotland and Ireland have responded with an open heart to the crisis, she added, saying that both countries are making the case for a co-ordinated European response as to do otherwise would be to neglect their past.
Ms Sturgeon said that the two countries share more than 1,000 years of history, and relations are warmer than they have been in the past.
She added: "The Scottish government knows and understands how important it is to maintain an open border within Ireland" and indicated that the Scottish government will support that.
As it happened: Nicola Sturgeon addresses Seanad
Nicola Sturgeon: 'The Scottish government knows and understands how important it is to maintain an open border within Ireland' pic.twitter.com/VilpZNvbMO
— RTÉ News (@rtenews) November 29, 2016
Ms Sturgeon spoke of the cultural, political and economic links between the two nations and recalled remarks of former president Mary Robinson where she attributed the revival of Irish culture to European membership.
She said Scotland's experience of the EU has not been quite the same as "we are not an independent member state."
She said that Scotland voted to remain in the EU as being European "has become a positive part of who we are".
She said that Scotland believes the UK should remain part of the European Single Market.
To guard against the possibility of the UK removing itself from the single market, the Scottish government will publish proposals for Scotland within the UK.
Ms Sturgeon added that there is also the option, "which remains firmly on the table," of Scotland considering again its future within the UK.
Earlier, Ms Sturgeon said the UK government does not appear to have reached the point where it has a clear negotiating strategy for Brexit.
She said that was partly because there were deep divisions in the Conservative party.
But she added that it was very frustrating for those who, like her, were anxious about the situation that lies ahead.
Opening proceedings in the Seanad this morning, Cathaoirleach Denis O'Donovan said: "This is a historic occasion which will strengthen and deepen the roots between our two countries."
He spoke of the the ties between the two countries through history and said that relations between the two countries were extremely close.
Senator O'Donovan said that he hopes the First Minister can secure the optimum outcome for Scotland in the Brexit negotiations.
The second day of Ms Sturgeon's Irish visit got under way with a business breakfast organised by employers' group Ibec.
She addressed around 130 business leaders on topics including building better business relationships between Ireland and Scotland.
Ms Sturgeon told the gathering that "Scotland is open for business."
Scotland will do everything possible to mitigate the impact of Brexit on Ireland, she added, saying that maintaining an open border is central to her strategy for a soft Brexit.
Protecting Scotland's interests in the EU post-Brexit will be a dominant theme during Ms Sturgeon's visit.
.@NicolaSturgeon says there is huge potential for a "Celtic corridor" to include Rep Irl, Northern Irl and Scotland #brexit @ibec_irl pic.twitter.com/28XUlROKlD
— Sharon Tobin (@sharontobin) November 29, 2016
After meeting Minister for Foreign Affairs Charlie Flanagan yesterday, the First Minister said she wanted the UK as a whole to retain membership of the single market, and that it would be welcomed in Ireland.
Ms Sturgeon also said that Scotland would support efforts to resolve any post-Brexit issues regarding the North-South border.
'No planning done for a Brexit vote', says Sturgeon
Speaking on RTÉ's Morning Ireland, Ms Sturgeon said: "There was no planning done for a Brexit vote and in the five months since, the UK government appears to have done very little to get to the position where it has a clear negotiating strategy.
"Hopefully that will change in the months that lie ahead, but it is frustrating at the moment for all of us, who are anxious about the situation that lies ahead."
Ms Sturgeon said that Taoiseach Enda Kenny has been extremely supportive of Scotland, but added that she understood his top priorities would be "all-Ireland issues".
She said she hoped that a consensus could be reached on issues surrounding the Good Friday Agreement and the peace process.
Ms Sturgeon said she felt that not enough importance was given to these issues during the referendum campaign.
She said she wanted the UK to stay in the single market, saying that leaving it would have a significant negative impact on jobs and the economy.
Should the UK government decide to leave the single market, she said, then she wanted to explore options for Scotland to remain in the market.
Despite the different positions of UK politicians during the referendum, there was now, she said, a need to hear all the voices from the devolved governments.
If Scotland is forced to leave the EU with damaging consequences for jobs, investment and living standards, then Scotland should have the right to decide whether that is the direction it wanted to take, she said.
This could mean that a second independence referendum might be held, she said, but she refused to be drawn on a possible date for a referendum.
She added that she would be engaging with the ongoing Brexit process in good faith.
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