This Was David Letterman ONE MONTH Before He ABANDONED NBC For CBS (Aired on May 20, 1993) 

Letterman, who had hoped to get the hosting job of The Tonight Show following Johnny Carson's retirement, moved to CBS in 1993 when the job was given to Jay Leno. This was done against the wishes of Carson, who had always seen Letterman as his rightful successor, according to CBS senior vice president Peter Lassally, a one-time producer for both men.[18] Letterman announced the move on January 14, 1993. On April 25, 1993, Lorne Michaels chose Conan O'Brien, who was a writer for The Simpsons at the time and a former writer for Michaels at Saturday Night Live, to fill Letterman's old seat directly after The Tonight Show.[20] O'Brien began hosting a new show in Letterman's old timeslot, taking over the Late Night name on September 13, 1993.
When Letterman left, NBC asserted their intellectual property rights to several of the most popular Late Night segments. Letterman easily adapted to these restrictions for his CBS show: The "Viewer Mail" segment was continued under the name "CBS Mailbag," and Late Night fixture Larry "Bud" Melman continued his antics under his real name, Calvert DeForest. Similarly, the in-house band (now free to add horns) was unable to use the name "The World's Most Dangerous Band," so the name was changed to "Paul Shaffer and the CBS Orchestra". The name "CBS Orchestra", approved by CBS (who retained rights to the name after Letterman retired in 2015), was Shaffer's idea. Notably, however, "Stupid Pet Tricks" originated on Letterman's 1980 early morning show The David Letterman Show, to which Letterman, not NBC, owned the rights. This meant "Stupid Pet Tricks" was able to cross over to the CBS show with its name and concept unchanged. With Carson retired, Letterman was also granted free use of some of Carson's sketches, and in due time, "Stump the Band" and "Carnac the Magnificent" (with Shaffer as Carnac) entered the Late Show rotation.