Jon Stewart’s announcement that he will give up his anchor chair on Comedy Central’s “The Daily Show” by the end of the year reverberated on Wednesday through the corridors of power in Washington, New York publishing houses and most important, the network’s parent company, Viacom.

Mr. Stewart’s departure, after a 16-year run skewering the foibles of a self-impressed news media and political class, presents Viacom with the delicate decision of finding a replacement for a show that, along with “South Park,” defines Comedy Central. And it means the cable network will enter the 2016 election year without two of its signature late-night personalities — Mr. Stewart and Stephen Colbert, who is taking his talents to CBS.

Viacom executives learned only recently that Mr. Stewart had definitively decided to leave the show he joined in 1999 and they have not settled on a successor. But they understand the need to tread carefully so as not to disturb the show’s successful concoction of current events and pointed satire.

“We’re not interested in doing a celebrity show,” said Doug Herzog, president of the Viacom Entertainment Group.