As a young producer at CNN in the 1990s, Sam Feist spent countless hours working on “Crossfire,” one of the first cable news shows to pit partisan pundits against one another. At lunch one day, the co-host Michael Kinsley mused about an alternative idea: “Ceasefire,” a program where Republicans and Democrats tried to find areas of agreement.

“It sat with me for, gosh, 20-something years,” Mr. Feist recalled.

Now Mr. Feist is the chief executive of C-SPAN, the low-key public affairs network beloved by political junkies. And “Ceasefire” is about to become a reality.

Envisioned as a respectful conversation between lawmakers from opposite sides of the aisle, “Ceasefire,” which is expected to debut in the fall, will be C-SPAN’s first new weekly program in two decades. “No shouting, no fighting, no acrimony,” Mr. Feist said in an interview. “Just two American political leaders with a willingness to find common ground.”

And where, pray tell, does he expect to find those?

Mr. Feist, a fixture of the Washington press corps who led CNN’s elections coverage for many years, acknowledged with a laugh that bipartisan relations in the nation’s capital were at a low ebb. That, he explained, is why a show like “Ceasefire” is sorely needed.

“The country rarely sees Republicans and Democrats engaged in a productive conversation,” he said. So for the past year, every time he has met with a member of Congress, Mr. Feist has pitched his idea for the show and asked the lawmaker who his or her best friend from the opposing party is.