Their rep says the birth went down Tuesday morning ... and the lil Clooneys are named Ella and Alexander.
The high-profile couple had actually been keeping a low-profile lately in England. The last time Amal was out in public was at "The Promise" premiere on April 6 in London. You'll recall, George showed up to an event 20 days later in London ... with no Amal in sight.
Amal did throw George a surprise birthday party last month in London.
Their reps says mom and the kids are "healthy, happy and doing fine. George is sedated and should recover in a few days."
More like 18 years. Congrats!