"The virus has not gone on holiday and neither have we," the premier said on a visit to the northeastern city of Lille.
"We need to protect ourselves against this virus, without putting a stop to our economic and social life, in other words avoiding the risk of a new generalized lockdown."
The country has emerged relatively swiftly from a two-month lockdown imposed to combat the virus, but this has come with the risk of an increase in cases.
France, which has registered over 30,000 deaths from the COVID-19 epidemic, recorded thousands of new confirmed infections last week, prompting some regions to reimpose local restrictions.
"We are seeing an increase in the figures for the epidemic which should make us more attentive than ever," Castex said.
"I call on every French person to remain very vigilant. The fight against the virus depends of course on the state, local communities, institutions, but also on each of us," he added.
Castex, a right-wing technocrat, was named prime minister last month in a cabinet reshuffle by President Emmanuel Macron.
Before his appointment, Castex masterminded the plan that took France out of lockdown.--AFP