NEW YORK — CBS moved swiftly Thursday to replace the retiring David Letterman with Comedy Central’s Stephen Colbert, who will take over the “Late Show” next year.

Colbert, 49, has been hosting “The Colbert Report” at 11:30 p.m. since 2005, in character as a fictional conservative talk-show host.

“Simply being a guest on David Letterman’s show has been a highlight of my career,” Colbert said. “I never dreamed that I would follow in his footsteps, though everyone in late night follows Dave’s lead.”

Mayors of New York and Los Angeles have already publicly urged the new “Late Show” host to choose their city. New York would appear to have the edge, since Colbert is already based in New York and CBS owns the Ed Sullivan Theater.