Weisz's publicist has confirmed the actress' casting to TheWrap, while Affleck's representatives did not immediately return TheWrap's messages.
The project was announced at the Berlin Film Festival, where it was described as a "romantic drama" and a "powerful and moving love story."
Production is scheduled to begin the first week of October in Oklahoma.
Affleck reportedly told a store employee that he was filming a movie in Bartlesville, and would be playing a fisherman.
The head of Oklahoma's Film & Music office would not confirm that Malick would be shooting a movie in the state.
Affleck is represented by WME, while Weisz is represented by CAA.