The James Bond actor, 50, and Oscar-winning The Constant Gardener actor, 48, are said to have told friends they are “very happy”.
The couple had been friends for many years before working together in 2011 film Dream House.
They began dating in 2010 and married in June 2011 in a private New York ceremony, with just four guests in attendance.
Weisz first publicly acknowledged her pregnancy in April. Speaking to The New York Times she said: “Daniel and I are so happy. We’re going to have a little human. We can’t wait to meet him or her. It’s all such a mystery.”
Both have one child each from previous relationships. Weisz has a 12-year-old son with Black Swan director Darren Aronofsky and Craig has a 26-year-old daughter from his marriage to actress Fiona Loudon.
The couple are well known for keeping their relationship extremely private.
In an interview with ES magazine earlier this year, Weisz said: “It’s very personal, it’s very private. I don’t think mine’s particularly exceptional apart from that we’re both in the public eye.”
It is unclear where the couple will raise their daughter. In an appearance on The Late Show with Stephen Colbert, Weisz was asked “Is the child going to be raised American or English?”
“I suppose it will be bilingual!” she said.
Weisz’ most recent onscreen performance is in period film The Favourite, which premiered at Venice Film Festival last week.
Craig will star in Bond 25, the working title of the next in the instalment, filming of which was due to begin in December but has been delayed due to the departure of director Danny Boyle.
It is rumoured to be his final appearance in a 007 role.
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