That is the finding of a survey carried out by the pollsters, Ipsos Mori, for BBC Northern Ireland's political programme, The View.
In the wake of June's EU referendum result, Sinn Féin demanded that the secretary of state should call a border poll as provided for under the Good Friday Agreement.
The Department of Foreign Affairs in Dublin also reported a sharp rise in the number of people from Northern Ireland applying for Irish passports.
However, this survey reveals that less than a fifth of the people interviewed by Ipsos Mori say the Brexit decision has influenced how they would vote in a referendum to decide whether Northern Ireland remains in the UK or joins the Irish Republic.
More than 1,000 people were interviewed by the pollsters face to face at locations across Northern Ireland between mid-August and early September.
A third of those interviewed (33%) want the government to call a border referendum.
However, more than a half (52%) oppose such a move. Opposition was strongest among those with a Protestant background with 72% against holding a border poll.
If such a poll is held, a clear majority of people, 63%, say they will vote for Northern Ireland to remain in the UK, while only 22% would support a United Ireland.
Some 13% of those polled don't know how they would vote, whilst another 2% would not participate in a border poll.
The same question was asked for the BBC by Ipsos Mori three years ago.
A direct comparison with that survey shows a slight decrease in support for staying in the UK, estimated at 65% in 2013, two points higher than the latest figure -  which is within the margin of error so not statistically significant.
Ipsos Mori reported a 5 point increase in support for a United Ireland (22%), which was just 17% in the 2013 survey - an increase which is regarded as a significant change.
The overwhelming majority of those with a Protestant background (88%) would vote to stay in the UK.
More than third of those with a Catholic background (37%) would also opt to stay in the UK, similar to the 2013 figure of 38%.
More than four out of 10 people with a Catholic background (43%) would back a United Ireland, up 8% on the 2013 figure of 35% - an increase which is regarded as statistically significant.
People were asked whether the result of June's EU referendum had changed their views on Northern Ireland's constitutional future.
Some 83% said the Brexit decision had not altered their position, while 17% indicated it had changed their thinking.
Those whose views had been influenced by the EU result were slightly more likely to be female, from a Catholic background and drawn from the affluent AB social classes.
In the immediate aftermath of the EU referendum, the Scottish First Minister Nicola Sturgeon said that a second Scottish independence vote was "highly likely".
Earlier this month, Ms Sturgeon asked SNP activists to take the lead in what she described as a "new conversation" on independence.
In the past, the Sinn Féin President Gerry Adams has predicted that developments in Scotland could have "seismic implications" for politics in Ireland.
In the Ipsos Mori survey, people were asked whether a future vote for independence in Scotland would change their view on a border poll.
A majority (56%) reckoned a decision by Scotland to break from the UK would not alter the way they would cast their vote in such a border poll.
Some 18% told Ipsos Mori that Scottish independence would make them more likely to vote for Northern Ireland to remain in the UK.
But 15% believed Scottish independence would make them more likely to vote for a United Ireland.
Under the Good Friday Agreement, the Northern Ireland Secretary James Brokenshire is supposed to call a border poll if it appears "likely to him that a majority of those voting would express a wish that Northern Ireland should cease to be part of the United Kingdom and form part of a united Ireland".
The Northern Ireland Office has previously ruled out calling such a poll.
Although the Ipsos Mori survey suggests a slight increase in support for a united Ireland, given that less than a quarter of the people of Northern Ireland would vote for a change, it seems unlikely that the UK government will shift its position on calling a border poll in the near future.
To view the full survey, click here.
More than eight out of 10 people in Northern Ireland say the UK's decision to leave the European Union has not changed the way they would vote in a future border poll.