The policy shift is aimed at being a model for the Catholic Church worldwide.
The mandatory reporting provision marks the first time the Vatican has put into law requirements for Catholic officials to report allegations of sex crimes to police or face fines and possible jail time.
Francis also issued child protection guidelines for Vatican City State and its youth seminary, acting after the global sex abuse scandal exploded again last year and the Associated Press reported that the headquarters of the Catholic Church had no policy to protect children from predator priests.
The law for the first time provides an explicit Vatican definition for "vulnerable people" who are entitled to the same protections as minors under church law.
The Vatican amended its canon law covering sex abuse to include "vulnerable adults" several years ago, but never defined it.
According to the new Vatican definition, a vulnerable person is anyone who is sick or suffering from a physical or psychiatric deficiency, is not able to exercise personal freedom even on occasion and has a limited capacity to understand or resist the crime.
The new law covers all personnel who live and work in the Vatican, the 110-acre city state in the centre of Rome, as well as the Holy See's vast diplomatic corps in embassies around the world.
The Vatican's ambassadors have figured in some of the most scandalous cases of abuse in recent years, with papal representatives accused of groping, distributing child pornography and sexually abusing minors in far-flung posts.
The law now requires any Vatican public official who learns of an allegation of abuse to report it to Vatican prosecutors "without delay".
Failure to do so can result in a fine of up to 5,000 euros (£4,300) or, in the case of a Vatican gendarme, up to six months of prison.
The legislation requires that victims be welcomed, listened to and provided with medical, psychological and legal assistance, and sets the statute of limitations at 20 years past the victim's 18th birthday.
Victims and their families are to be protected from any retaliation, the law says, answering long-standing problem faced by victims, including one who reported sexual abuse at the Vatican's own seminary only to be kicked out the following year.
Mimicking some provisions in place in the US church, background checks will be required for staff and volunteers working with minors and there will be safe environment training for all Vatican personnel.
The accused is to be removed from their job pending the investigation, and be allowed to defend themselves.
The Vatican's editorial director, Andrea Tornielli, said that while few children live in the Vatican City State, Francis decided to make the legislation and accompanying guidelines a model.
Last year, AP reported that Vatican City had no policy to protect children or require suspected abuse to be reported to police, even though the Holy See required such policies in Catholic dioceses around the globe and had told the UN in 2013 that such a policy was in the works.
The absence of clear-cut policy became evident following revelations that a teenage seminarian in the Vatican's youth seminary had, in 2012, accused one of the older boys of molesting his room-mate.
Nothing came of it. Vatican police, who have jurisdiction over the territory, were not called in to investigate. A series of bishops - including Cardinal Angelo Comastri, Francis's vicar for Rome and the archpriest of St Peter's Basilica - said they investigated, but no one ever interviewed the alleged victim.
The student who lodged the complaint, Kamil Jarzembowski, was promptly kicked out of the seminary while the accused seminarian was ordained as a priest.
On Friday, Mr Jarzembowski said the law was a positive step forward, particularly its explicit recognition that the pre-seminary falls under its jurisdiction.
Last December, the Catholic Church vowed "never again" to cover up sexual abuse perpetrated by clergy.
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