The 32-year-old Mason played in a career-high 51 games for the Predators last season, posting a 2.90 goals-against average and a .898 save percentage.
"Acquiring Chris really solidifies our goaltending position which was one of our goals going into this weekend," Blues president John Davidson said in a statement. "We are looking forward to a big day today at the draft."
Nashville has two picks each in the first, second and fourth rounds. The team also agreed with goalie Dan Ellis on a multiyear deal.
Mason compiled a 58-43-12 record in 135 appearances for Nashville during two stints with the team (1998-01 and 2003-08), mostly serving as a backup behind former goalie Tomas Vokoun and Ellis.