The far right made significant gains but finished third, behind President Emmanuel Macron's coalition and well below expectations.
The far-right Rassemblement National and its allies were projected to win between 138 and 145 seats, and Macron's coalition, Ensemble, between 152 and 158.
That leaves Macron in a position to negotiate with parties that have long been in opposition to him to try to form a coalition government.
"The president has the duty to call on the Nouveau Front Populaire to govern," declared left-wing leader Jean-Luc Mélenchon.
Mélenchon praised the electorate's efforts to obtain "a result that was said to be impossible" – a victory for the left.
Bardella spoke of "unnatural alliances" between the presidential coalition and the left, which he said had led to his party's defeat.
The previous record for the far right, set two years ago, was 89 seats. »