The Mirror reports that the pair took home prizes for best acting in a play, with Law winning for Hamlet and Weisz for her portrayal of Blanche DuBois in A Streetcar Named Desire.
Speaking on a video message, Law said that the award was a "real thrill" and added: "It's a very important award for me because it's voted for by the theatregoers. Thank you very, very much. And I look forward to seeing you all again soon, on the stage."
Elsewhere, former X Factor contestant Diana Vickers took home the newcomer of the year award for her role in Little Voice, while Sir Patrick Stewart and Sir Ian McKellen shared the theatre event of the year for Waiting For Godot.
Stewart also won best supporting actor for his role in the Royal Shakespeare Company's production of Hamlet.
Oliver! also did well at the awards, with Rowan Atkinson winning best actor in a musical and I'd Do Anything star Jodie Prenger being honoured as best supporting actress in a musical for her portrayal of Nancy.