Antoine Armand, center, deputy for the Savoie region, in the French Alps, speaks at the National Assembly, Thursday, June 8, 2023 in Paris.AP
The suspect was arrested at the scene in the town of Annecy after the attack at around 9:45 am (0745 GMT).
He told police he was a Syrian asylum seeker, a police source told AFP on condition of anonymity.
Interior Minister Gerald Darmanin tweeted that he had been arrested "thanks to the rapid reaction of security forces".
Two of the children -- believed to be aged around three -- and the adult were in critical condition and fighting for their lives in hospital, a security source told AFP.
Local officials and a security source had previously said six children and another person were hurt.
French President Emmanuel Macron denounced the stabbing as an act "of absolute cowardice."
"The nation is shocked. Our thoughts are with them as well as their families and the emergency services," he wrote on Twitter.
Minute of silence
Prime Minister Elisabeth Borne's office announced she was travelling to the scene and MPs in the national parliament held a minute's silence as news of the attack filtered through in the French media.
"There are very young children in a critical state and I ask you to observe a minute of silence for them, for their families," parliament speaker Yael Braun-Pivet told MPs as she interrupted a raucous debate about pension reform.
"We hope that the consequences of this extremely serious attack are not consequences that will send the country into mourning," she added.
The identity of the attacker was being checked by security forces and has not been confirmed.
France has been the target of a series of traumatic Islamist attacks over the last decade, including on the Charlie Hebdo magazine in 2015, the national stadium and Bataclan concert hall in Paris in the same year, and the city of Nice in 2016.
Most recently, the beheading of a teacher in broad daylight in 2020 near his school in a Paris suburb by a Chechen refugee led to an outpouring of grief and a national debate about the influence of radical Islam in deprived areas of the country.
Annecy is a quiet Alpine town of 135,000 people that lies 30 kilometres (20 miles) south of the Swiss city of Geneva.