The campaign's spokesman John McGuirk told national broadcaster RTE the people of Ireland had "weighed it in the balance and it came down on one side."
"I obviously would have preferred if they had come down on the other," he said.
He added: "There is no prospect of the (abortion rights) legislation not being passed".
Irish Times reporter Sarah Bardon said the spokesman told her he had "made peace" with the exit polls.
His concession comes after Ireland’s 3.2 million voters were asked whether to scrap the eighth amendment of the Irish constitution which places the life of a mother and her baby on equal footing and effectively bans abortion.
The amendment was introduced via a referendum in 1983.
The Irish government will be able to legislate on allowing terminations if the results are found to be correct.
The historic referendum vote will be decided by a margin of 68 per cent to 32 per cent in favour of Yes, an Ipsos/MRBI survey predicted. An RTE exit poll put the Yes campaign’s lead even higher, at 69.4 per cent.
"The unborn child no longer has a right to life recognised by the Irish state. Shortly, legislation will be introduced that will allow babies to be killed in our country. We will oppose that legislation," Mr McGuirk said in a statement.
Ireland celebrates historic Yes vote on abortion referendum
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"If and when abortion clinics are opened in Ireland, because of the inability of the Government to keep their promise about a GP led service, we will oppose that as well.
"Every time an unborn child has his or her life ended inIreland, we will oppose that, and make our voices known."
Vote counting started at 9am on Saturday and an official result is expected to be announced in the late afternoon.
Taoiseach Leo Varadkar, who supported the campaign to liberalise Ireland’s abortion, said: “It’s looking like we will make history tomorrow".
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