Carney is the lead singer of the rock band Carney. He also appeared in the film "Snow Falling on Cedars."
"Having recently finished shooting 'The Tempest' with Reeve as the young prince, I am thrilled to cast him in an entirely different role as Peter Parker in 'Spider-Man,'" director Julie Taymor said in a statement. "When I first saw Reeve, he was performing with his terrific band Carney. The music of Bono and Edge requires an extraordinary talent. Not only is Reeve a great rock 'n' roll singer, but his acting is equally fresh and authentic."
He joins previously announced cast members Evan Rachel Wood as Mary Jane Watson, Peter's girlfriend, and Alan Cumming as Norman Osborn, also known as the Green Goblin.
The full producing team for the musical was also announced Friday. It includes Michael Cohl, Jeremiah J. Harris, Hello Entertainment's David Garfinkle, Marvel Entertainment's David Maisel and Sony Pictures Entertainment.
The producers said full financing for "Spider-Man" is expected to be in place shortly and, once this happens, the musical will resume full production.
The show is scheduled to open in 2010 at the Hilton Theatre in New York.
"Having started early on with 'Spider-Man' and now taking on the lead producing role is extremely exciting and a welcome challenge," Cohl said in a statement. "Knowing the details and intricacies of this wildly ambitious project, I want to 'turn off the dark' on all the wild speculation about the show -- it's moving forward!"
"With a team that includes Bono, Edge and Glen Berger, there was never a question in my mind that 'Spider-Man' would happen -- it was just a matter of when," added Taymor. "I am delighted that Michael Cohl is making this a reality."