The National Assembly unanimously voted to amend the law, drawn up by Herve Morin when he was the defence minister in 2009.
Most claims for compensation for ill health because of the weapons tests have been rejected which prompted sustained calls for the law to be amended.
Mr Fritch said the Assembly decision marks a historic day and a relief.
He has hailed the territory's parliamentarians in Paris for their work and praised the tireless efforts of the nuclear test veterans organisations and the Maohi Protestant Church.