The man shot the job centre employee after he entered the office in the city of Valence, said a police source, who asked not to be identified by name, confirming a report first published in the Dauphine Libere newspaper.
The assailant then went to the nearby town of Guilherand-Granges where he shot twice a female employee of a refuse collection company.
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She later died of her wounds, Valence prosecutor Alex Perrin told AFP.
The suspected gunman fled by car towards the centre of Valence where his vehicle struck a police car trying to stop him. He was then arrested, the police source said.
Perrin confirmed the man, whose motives were not immediately clear, was now in detention.
Prime Minister Jean Castex said the events had put “the whole country into mourning” while Labour Minister Elisabeth Borne was due at the scene.