DUBLIN—As Ireland prepares to welcome its first papal visit in four decades when Pope Francis arrives Saturday, it is a much changed and less Catholic country. But in one key respect, the Ireland of 2018 is as it was in 1979: The vast majority of its children are educated in Catholic schools.

In the years since millions came out to see Pope John Paul II, contraception and divorce have become legal, as has same-sex marriage, while a ban on abortion added to the constitution in 1983 was removed after a referendum this May.

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