WASHINGTON—During breaks in the Senate impeachment trial proceedings in the past week, a bevy of Republicans eager to defend President Trump have trooped through the hallways of the Capitol to address the media, sometimes waiting in line to make their case.

Sen. Ted Cruz (R., Texas) has taken to another venue: the basement podcast studio in a nondescript building in downtown D.C.

Each night after the trial wraps up, Mr. Cruz has arrived at the studio to record the latest episode of a new podcast, “Verdict with Ted Cruz.” The show, which has soared to the top of the charts since its launch last week, features Mr. Cruz and conservative media personality Michael Knowles firing broadsides at the impeachment case and, often, former Vice President Joe Biden.

The podcast has been downloaded more than a million times.

As much as Senate Republicans have pushed for a timely conclusion to the third-ever impeachment trial in U.S. history, the success of Mr. Cruz’s podcast has demonstrated the possible upside of the vitriolic politics of the process.