A surge in Green Party support is anticipated following the Republic of Ireland's local elections, as vote counting gets under way on Saturday.
The government's proposal to liberalise the country's divorce laws is also expected to be carried by an overwhelming majority.
The government wants parliament to decide how long a couple should be separated before divorce is allowed.
Counting for the European Parliament will begin on Sunday.
An RTÉ/TG4 exit poll published on Friday night suggests there has been a surge in support for the Greens in the local elections but the party will still be behind the governing party Fine Gael, the main opposition party Fianna Fáil and Sinn Féin.
The poll indicates that Fine Gael and Fianna Fáil are both on 23%, Sinn Féin on 12% and the Greens on 9%.
The exit poll also suggested 87% support for the divorce referendum proposal.
That finding is in line with the only opinion poll carried out for the Irish Times during the campaign.
At the moment, someone can only apply for a divorce if he or she has been separated for four of the previous five years.
In 1995, the Republic voted to allow divorce by a tiny margin, with a backing of 50.3%.
In the referendum, voters were asked to remove time frames from the constitution and to allow the Irish parliament to decide the issue.
The government has said it believes a two-year separation period is long enough.
There was no organised campaign against what was planned.
Approval of the referendum to amend the constitution would be no surprise coming after recent referendums that showed big majorities for same-sex marriage and abortion rights.
Counting in Waterford, Cork and Limerick on a proposal for directly elected mayors with enhanced powers will begin on Monday.