“I will never compromise when our freedom is at stake,” said Ms. Cheney, a not-so-subtle reference to Mr. Enzi’s penchant for working with Democrats to find common ground.

Word of Ms. Cheney’s announcement went out just minutes after Mr. Enzi, in a barely veiled message to Ms. Cheney, issued his own statement that he would seek a fourth term. “I intend to run for re-election for Wyoming’s Senate seat in 2014,” Mr. Enzi said. “When I announce formally, I will let everyone know that date in the future. In the meantime, I will do the job I was already elected to do. My trips to Wyoming almost weekly, the public listening sessions, the groups I meet with and speak to, working behind the scenes — this is what I have been doing since I was elected, and this is what needs to be done.”

Talking to reporters in the Capitol after the video went public, Mr. Enzi said he was not notified by either Ms. Cheney or her father — whom he has known for over 30 years — about Ms. Cheney’s intentions.

“I thought we were friends,” he added.

Mr. Enzi said in an interview this month that he was planning on running again but that he wanted to delay any declaration until closer to the election. Yet he also revealed at the time that Ms. Cheney had called him to say she was considering challenging him.

Her ambitions had been rumored since she moved from suburban Washington to the Jackson Hole, Wyo., area last fall and immediately became a regular at state political and civic events. Ms. Cheney, a former senior State Department official in President George W. Bush’s administration, has been a Fox News contributor.