Tensions between the two neighbours have escalated since Italy's new government barred the Ocean Viking ship from docking with 230 migrants.
French Interior Minister Gérald Darmanin denounced Italy's "unacceptable behaviour".
He said the boat would be allowed to dock on Friday in the port of Toulon.
Italy did not respond immediately to the announcement, but the charity that runs the boat, SOS Méditerranée, said it was both relieved by the French decision and angry that for three weeks those on board had been let down by Europe's "dramatic failure".
Mr Darmanin warned of "extremely severe consequences for our bilateral relations" with Italy. For a start, France would not take in 3,500 refugees it had earlier agreed with Italy to accept in protest at Rome's actions. He also said France would take measures to strengthen controls on the border with Italy.
Italy's right-wing government came to power last month under Giorgia Meloni, who before she became prime minister vowed to blockade migrant boats trying to leave North Africa for Italy.
The Ocean Viking was one of four charity ships that sailed for Italy with a total of more than 500 migrants rescued in the Mediterranean.
Those on board the other three ships were eventually allowed to disembark at Italian ports, but only after initial refusals. At one point three of those on board the Geo Barents rescue boat leapt into the water to reach land.
Ocean Viking was denied entry into port and SOS Méditerranée said its 43 requests to Italian authorities met with no response.
Who is responsible for helping migrants at sea?
It sailed on towards Corsica and four people on board were airlifted to hospital on Thursday morning. SOS Méditerranée, cited by AFP, said one of the migrants on the ship was in an unstable condition and had not reacted to treatment since late October.
The French interior minister then announced the others would be allowed to leave the ship at the naval base in Toulon on an "exceptional" basis.
He went on to criticise Rome's "reprehensible" attitude and stressed the migrants had been picked up in Italy's search and rescue zone in the Mediterranean.
"We regret Italy has failed to meet its duty of humanity," Mr Darmanin told a press conference, accusing France's neighbour of not conducting itself as a "responsible European state".
A third of passengers would be "relocated" to France, he said, while another third would be sent to Germany and the others shared out between EU member states.
The European Commission said on Wednesday that the ship should be able to immediately disembark at the nearest place of safety.
In a statement, it did not mention Italy or France by name, but said there was a "clear and unequivocal" legal obligation to rescue distressed persons at sea.
French far-right leader Marine Le Pen reacted angrily to France's decision to allow the Ocean Viking to disembark in Toulon, accusing President Emmanuel Macron of a dramatic show of lenience towards "massive and anarchic immigration".