Taoiseach Leo Varadkar has called on Independent TD Mattie McGrath to "cool the conspiracy theory" about illegal voting and illegal registration of voters.
Mr Varadkar also said claims that French students studying in Ireland had been added illegally to the electoral register had been refuted.
And he said it was “very encouraging to see a huge number of young people had registering to vote in the past couple of weeks”.
He said he was encouraged “that there has been such an upsurge, no matter how they vote, in the number of young people registered to vote”.
Mr Varadkar was speaking in the Dáil after Mr McGrath picked up on comments the Taoiseach had made in response to questions about when legislation on abortion would be introduced if the electorate voted to repeal the Eighth Amendment on Friday.
The Taoiseach told People Before Profit TD Bríd Smith that “I think we should wait until the people speak before we move to legislation”.
“There’s always the possibility of a legal challenge. That has happened before.”
He added: “With the exception of a few postal voters very few people have voted and I would caution against any complacency at this stage.”
Picking up on this Mr McGrath said: “Who else might have voted, I wonder.”
He said he had previously asked Minister for Local Government Eoghan Murphy “about the French students who unknown to themselves were put on the register here illegally at NUIG in Galway”.
The Tipperary TD claimed “the register is a mess and people have been put on and put on twice”.
He asked Mr Murphy “when are you going to have an investigation? When the votes are counted, is it?”
He called on Minister for Local Government Eoghan Murphy to carry out a check into illegal activity. “It is an undocumented mess, that’s what it is with people on it illegally.”
He told the Taoiseach: “You might look at that instead of campaigning for more and more Yes votes.”
Mr Varadkar replied: “If you’d cool the conspiracy theory Deputy McGrath - the people who’ve voted so far are postal voters, although there is early voting on some islands and also some nursing homes.”
The Taoiseach added: “The allegation that some French students were on the register - I understand that has been refuted.”
Last week a student from Brest in France, on a year-long Erasmus programme was reported to have received a polling card to the address she was staying in, in Galway, even though she did not apply to the local authority and was not legally entitled to do so.
Mr Varadkar said of the upsurge in the number of young people registering to vote: “I think that indicates that young people want to have a say in this referendum.
“And it is young people who are the people who are going to be most affected by laws that we make in relation to the termination of pregnancies.”