Mr Molins said 19-year-old Adel Kermiche was under house arrest and wearing an electronic monitoring bracelet at the time of the attack.
Kermiche and an unknown accomplice, armed with knives, stormed the church in Saint-Etienne-du-Rouvray on Tuesday, taking hostage 86-year-old priest Father Jacques Hamel, three nuns and two worshippers.
Police shot and killed the attackers as they emerged from the church, freeing three hostages, one of whom was seriously wounded.
The priest was found dead with his throat slit, and an 86-year-old worshipper had severe knife wounds.
Both attackers had on them a "fake explosive device covered in aluminium foil", Mr Molins said. Witnesses say they pledged their allegiance to the Islamic State group (IS) during the attack.
Kermiche, who was from the Normandy town where the attack took place, last attempted to reach Syria in May, 2015, and was detained until March but forced to wear an electronic tag to track his whereabouts.
He was allowed to only leave his home for a few hours a day, Mr Molins said. Police were still seeking to identify the second of the two attackers and raids were underway. Police said one person had been arrested.
News agency Amaq, which is affiliated with IS, said two of its "soldiers" had carried out the attack.
'Sermon in Arabic around the altar'
The knifemen arrived during morning mass in Saint-Etienne-du-Rouvray, near Rouen, north-west of Paris, where Father Hamel, the parish priest, was leading prayers.
"They forced him to his knees and he tried to defend himself and that's when the drama began," Sister Danielle, who escaped as the attackers slayed the priest, told RMC radio.
"They filmed themselves. It was like a sermon in Arabic around the altar," the nun said.
One of the nuns managed to escape and call police, who, upon arrival, tried to negotiate with the hostage-takers through a small door.
Mr Molins said police were unable to launch an assault on the church as three hostages were lined up in front of the door.
Two nuns and one worshipper then exited the church followed by the two attackers, one carrying a handgun, who charged police shouting "Allahu Akbar" (God is greatest) before being shot.
Speaking at the scene, French President Francois Hollande called it a "dreadful terrorist attack", adding that the attackers had pledged allegiance to IS, the militant group that he said had declared war on France.