Amal gave birth to the twins in London on Tuesday morning, reported People magazine.
"This morning Amal and George welcomed Ella and Alexander Clooney into their lives. Ella, Alexander and Amal are all healthy, happy and doing fine," the actor's publicist Stan Rosenfeld said in a statement on behalf of the couple.
"George is sedated and should recover in a few days," they jokingly added.
These are the first children for both Amal and George.
Amal's boss, Geoffrey Robertson QC of Doughty Street Chambers, is also elated for the couple, "All I can say is that it's a great co-production."