Abortion is banned in Ireland under its Eighth Amendment, which gives an unborn fetus equal rights. If the "yes" vote is confirmed, Ireland's parliament will be tasked with writing new regulations on abortion.
Irish voters "weighed it in the balance and it came down on one side".
The referendum vote is expected to be a close contest in the largely Catholic nation.
Those who want to retain the Eighth Amendment will vote no, while those who want to replace it will vote yes.
Abortion is now only legal in Ireland in cases where the mother's life is at risk, and remains illegal in cases of rape, incest or fetal abnormality.
Connacht-Ulster - the part of Ireland that is most rural and most conservative - reported 59 percent in favor of constitutional change. The procedure is banned, though, even in cases of rape, incest or fetal abnormality.
The 31-year-old was miscarrying her first baby and was refused a termination at a hospital in Galway.More news: Obamas sign deal with Netflix
Michelle Sweeney has lived in Vancouver for the past year and drained her savings account for a last minute ticket home to Ireland on the afternoon of May 22.
She didn't get the medical treatment she needed because of the Eighth Amendment, Halappanavar's father told the Guardian in an interview.
Some "Yes" politicians were already calling on the government to pass the legislation without delay following the publication of the exit polls. Getting an illegal abortion within Ireland can lead to a life sentence in prison.
Surprisingly, they also suggest that supporters of more liberal abortion laws may have triumphed throughout the country, not just in the cosmopolitan capital, Dublin, where a strong youth vote had been anticipated.
They correctly perceived that Ireland was gradually becoming a more secular country, and wanted to create a bulwark against the introduction of abortion that would be hard to reverse.
Are abortion rights at risk anywhere?
Currently, 78 percent of the Irish population is Catholic, and members of the church are hoping their members will vote "no".
In 2017, the Irish government convened a Citizens Assembly to consider evidence on abortion law reform.More news: Red Bull's engine deficit halved in Monaco, say drivers
However, if a majority votes to repeal it, abortion still will not automatically become legal.
In 1983, the eight amendment to the Irish constitution was ratified, which outlawed abortion in the country.
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Most European countries allow abortion on request up to 12 weeks.
In the United Kingdom, abortions can be carried out up to 24 weeks with the agreement of two doctors.
Doctors who perform an abortion outside the law could be imprisoned for up to 14 years.
"The Yes vote ends decades of stigma, shame and silence around abortion and unintended pregnancy".More news: 'Hero' teacher tackled gunman, student says: 'Most people would never dare'