It's after the resignation of Edouard Philippe who has been in the post for three years.
Mr Castex, 55, is relatively unknown in France, although he is a senior civil servant.
The new Prime Minister has been a key player in the government's response to the Covid-19 pandemic.
US says it struck Iranian drone command sites at the weekend
The US has stated on Sunday it conducted "self-defense strikes" on Iranian radar and drone control sites in Iran's Goruk and Qeshm Island at the weekend in what it said was a response to "aggressive" actions from Tehran.
US proposes new plan to ease Israel-Lebanon tensions amid fighting
US Secretary of State Marco Rubio has spoke with both Lebanese President Joseph Aoun and Israeli Prime Minister Benjamin Netanyahu on the diplomatic negotiations between Israel and Lebanon and has proposed a plan to allow for "gradual de-escalation", a US official said on Sunday.
At least two dead at Hanwha Aerospace fire in South Korea, official says
At least two people have died and two others were injured in a fire at a factory operated by South Korea's Hanwha Aerospace in the city of Daejeon, a fire official said on Monday.
Netanyahu orders deeper Israeli incursion into Lebanon
Israeli Prime Minister Benjamin Netanyahu said on Sunday he had ordered troops to move further into Lebanon in the battle against Hezbollah, despite a ceasefire announced more than six weeks ago.
Ebola recoveries bring signs of hope in DRC as suspected cases emerge outside Africa
Four nurses who were being treated for Ebola caused by the Bundibugyo strain of the virus have been discharged from a hospital in Bunia in the Democratic Republic of Congo after recovering from the disease, the World Health Organisation said on Sunday.
