Terms of the deal announced Friday were not disclosed.
The 37-year-old Madden has won three Stanley Cup titles in his career, including last season with the Chicago Blackhawks. He had 10 goals and 13 assists in 79 games for Chicago.
Adding depth at center behind captain Mikko Koivu has been a priority for the Wild for the last few seasons. The team had interest in Mike Modano, but he signed with Detroit. Koivu signed a seven-year, $47.25 million extension last month.
Madden has long been one of the top defensive forwards in the game and has made the playoffs in each of his 10 full years in the league.