French President Emmanuel Macron has rejected a resignation offer from the prime minister he appointed, saying instead that the government must "stay on task and act".
Elisabeth Borne was criticised by some commentators after Mr Macron's coalition lost its majority on Sunday.
Her future looked under threat after the result, which left the president needing to secure support from rivals.
He is set to meet his political opponents for rare talks on Tuesday.
But neither Marine Le Pen's far-right nor Jean-Luc Mélenchon's left-green alliance are keen to work with Mr Macron's centrist government, which is 44 seats shy of a majority and desperately trying to avoid political paralysis.
The Élysée palace said Ms Borne formally offered to resign on Tuesday morning, submitting a letter to Mr Macron - the head of state - who turned it down.