It’s almost time for NBC’s Late Night switcheroo to begin, and the dates are now official: Conan O’Brien’s last show will air February 20, Access Hollywood reports, with Jimmy Fallon taking over on March 2. Then, Jay Leno’s last Tonight Show will air May 29, with Conan moving in June 1. Leno, of course, will headline a new prime-time show this fall. (More Jay Leno stories.)