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.
8 Israeli airstrikes hit Syria's Abu al-Duhur airbase, state TV reports
Eight Israeli airstrikes have hit the Abu al-Duhur military airbase in northwestern Syria early on Tuesday, targeting the runway and storage facilities, Syria's state-run Ekhbariya said, citing a military source.
Two killed in high school shooting in southern Philippines
Two students were killed in a shooting at a high school in the southern Philippines on Tuesday, one of them the gunman, officials said, in the second such incident in the country in less than two months.
Pakistan top court orders ex-PM Imran Khan be moved to hospital from jail
The Supreme Court of Pakistan has ordered on Tuesday that former Prime Minister Imran Khan be moved to hospital from jail, his Pakistan Tehreek-e-Insaf party said, meeting a longstanding demand of his party and family, who had raised concerns about his health.
Russian missile attack kills 10 in Ukraine's Kharkiv region, governor says
A Russian missile attack has killed 10 people and injured a further eight in a village in Ukraine's northeastern Kharkiv region, the regional governor said on Tuesday.
Trump says Iran won't make deal he thinks is needed
President Donald Trump said on Monday Iran would not make the deal he thinks is needed to end a war with the US and should surrender as talks between the countries remained stalled.
