Stan Rosenfield, publicist for her actor husband George Clooney, confirmed the babies were born on Tuesday.
In a statement, he said: “This morning Amal and George welcomed Ella and Alexander Clooney into their lives.
“Ella, Alexander and Amal are all healthy, happy and doing fine. George is sedated and should recover in a few days.”
The twins were safely delivered at a hospital in London, where the couple have been living during the pregnancy.
George Clooney, who is 56, married 39-year-old Amal Alamuddin back in 2014 – and this is the first time either of them have had children.
Sky’s Entertainment Correspondent Katie Spencer said: “This is the man who was the eternal bachelor.
“Ten years ago, he was the one saying he was never going to get married, never going to have children, never going to settle down. Amal is the one who changed all that.”
The couple had kept a low profile during the pregnancy, but news they were expecting was confirmed back in February by the actor’s close friend Matt Damon.
People magazine reports that former President Barack Obama had visited the Clooneys in their countryside home near London on 27 May.
Last weekend, George cancelled a visit to a humanitarian event in Armenia, telling organisers: “If I came there and my wife had twins while I was there, I could never come home.”
Unsurprisingly, the births have been welcomed with countless messages on social media – and it was not long before “Ella and Alexander” was trending on Twitter.