After official results showed more than two-thirds of voters backed repealing the ban, Varadkar says he wants to make sure there are fewer crisis pregnancies and better sex education in schools going forward.
If the partial results hold up, the referendum would likely end the need for thousands of Irish women to travel overseas - mostly to neighboring Britain - for abortions they can't get at home.
As Ireland now prepares to legislate for abortion in certain circumstances, The Irish Times would like to hear your reaction to the outcome of the referendum, wherever you are in the world.
Ireland officially banned abortion in 1983, largely in response to 1973's United States Supreme Court Roe v. Wade ruling, which legalized abortion in the US. In practical terms, the amendment outlawed all abortions until 2014, when terminations in rare cases when a woman's life was at risk started being allowed.
Varadkar said he wanted the law in force by the end of the year and Health Minister Simon Harris told AFP that the cabinet would meet on Tuesday to approve the drafting of legislation.
People celebrate at Dublin Castle as the official results of the referendum on the 8th Amendment of the Irish Constitution are announced in favour of the yes vote.
"What Irish voters did yesterday is a tragedy of historic proportions", Save The 8th said. The strongest backing came from younger voters - the exit poll said the only age group in which a majority voted "no" were voters who are 65 or older.
Irish lawmakers are expected to bring up legislation allowing for abortions up to 12 weeks of pregnancy and even later in cases where the life of the mother is at risk.
Ailbhe Smyth, a veteran campaigner and co-director of Together4Yes, the national pro-repeal group, is one of those women.
The amendment was approved in 1983 and pro-repeal campaigners say that since then nearly 170,000 pregnant women travelled overseas to have terminations.
"I said in recent days that this was a once in a generation vote", he says.
A huge 66.4 per cent chose to repeal the controversial Eighth Amendment on Friday, after months of campaigning gripped the country. "Today, the AUL legal team is saddened that the people of Ireland have paved the way for abortion on demand in their country".
Ireland, once seen as one of the most socially conservative countries in Western Europe, is poised to end its highly restrictive abortion ban. "Having shared our stories and shared our experiences, if people had voted "no" what would that have meant for us?"
The law on abortion is enshrined in the country's constitution, which can be changed only by referendum.