If it failed to it risked a fine of up to €150,000 and a second of €300,000 if it still failed to act.
The move is the result of a year-long tussle with the top search engine and email provider.
The CNIL said Britain, Germany, Italy, the Netherlands and Spain were undertaking similar infringement procedures that could result in fines on Google of several million euro in total.
Britain is still considering whether its law has been breached and will write to Google soon with its findings.
"By the end of July, all the authorities within the (EU data protection) task force will have taken coercive action against Google," said CNIL President Isabelle Falque-Pierrotin.
Google, which has repeatedly said that its approach to user data respects European laws, said it would continue to work with the authorities in France and elsewhere.
The CNIL's move is the latest salvo in a fight between EU governments and Google that began in March 2012 and is seen by legal experts and policymakers as a test of Europe's ability to influence the behaviour of international Internet companies.
The scrutiny comes at a delicate time after revelations that the US National Security Agency has secretly gathered user data from nine big US internet companies, including Google, to track a person's movements and contacts over time.