President Jacques Chirac says France is determined to prevail in the face of widespread rioting that has gripped mostly African and Arab communities.
Mr Chirac spoke after chairing crisis talks following the 10th - and worst - night of violence, arson and looting.
He said it was an "absolute priority" to re-establish order. But soon after, fresh unrest erupted in several cities.
In Grigny, south of Paris, rioters fired shots at police, injuring 10 police officers, two of them seriously.
Violent attacks were also reported in Orleans, Rennes and Nantes.
Rioters burnt nearly 1,300 cars and more than 300 arrests were made on Saturday night from the cities of Nice to Strasbourg.
"The law must have the last word," Mr Chirac told reporters in his first public address on the violence.
Mr Chirac's comments were his first public address on the unrest
"The Republic is quite determined... to be stronger than those who want to sow violence or fear."
Mr Chirac promised arrest, trials and punishment for perpetrators.
But he also noted that "respect for all, justice and equal opportunity," were needed to end the unrest.
Mr Chirac has faced criticism from opposition politicians for not speaking publicly about the unrest since it began on 27 October. His only previous comments came through a spokesman.
Sunday's remarks came after talks with key ministers including Prime Minister Dominique de Villepin and Interior Minister Nicolas Sarkozy, at the presidential palace in Paris.
Correspondents say such security meetings are not usually made public, and its announcement is indicative of the gravity of the situation.
Mr de Villepin promised extra security where it was needed.
"We cannot accept any 'no-go' areas", he said.
Unrest began after the deaths of two youths in the rundown Paris suburb of Clichy-sous-Bois, who were accidentally electrocuted at an electricity sub-station after reportedly fleeing police.
The northern town of Evreux in Normandy saw some of the worst unrest on Saturday night with at least 30 cars burned along with three shops, the local authorities said.
A school was also petrol-bombed in the town while four police officers were injured in clashes with youths, some of them reportedly wielding baseball bats.
In other incidents:
- A McDonald's was rammed by a car and almost completely burnt out in Corbeil-Essonnes, south of Paris
- Five classrooms of a nursery in Grigny were destroyed by fire while a primary school was also slightly damaged
- A recycling facility was attacked in Essone, with at least 35 vehicles torched
- In Drancy, north-east of Paris, two teenagers were caught and handed over to police after they tried to set fire to a lorry.
Clichy-sous-Bois: Two teenagers die in electricity sub-station on 27 October. Successive nights of rioting follow rumours they were fleeing police. A number of people arrested or injured.
Aulnay-sous-Bois: A flashpoint after violence spread from Clichy. Shots fired at police and cars and shops set ablaze. Further trouble in nearby suburbs, with more shots fired at police.
Elsewhere in France: From 3 November, violence spreads to other major cities including Dijon, Marseille, Nice and Strasbourg.