It was December 1992, and Mr. Bush, who had been defeated by Bill Clinton, was on his way out of the White House. He summoned his staff to the East Room for a formal Christmas greeting. But when "Hail to the Chief" began to play, it was not Mr. Bush who entered the room, but Mr. Carvey.

The crowd roared in surprise.  [...]

When the president took the lectern -- "I don't dare move my hands," he said -- he thanked Mr. Carvey for visiting the White House.

"Dana has given me a lot of laughs," Mr. Bush said. "He said to me on the phone, 'Are you sure you really want me to come there?' And I said, 'Yeah.' He said, 'I hope I've never crossed the line.' I knew exactly what he meant and as far as I'm concerned, he never has."

"The fact that we can laugh at each other," Mr. Bush said, "is a very fundamental thing."

In his life after the White House, Mr. Bush continued to embrace the comedian's impression of him, even referencing it in his eulogy to former President Gerald R. Ford in 2007. He also stayed in touch with Mr. Carvey over the years.

In an interview this year, Mr. Carvey told Conan O'Brien, the late-night host, that Mr. Bush wrote him notes at important moments in Mr. Carvey's life and even called him on Election Day in 2004. "We had so many warm moments with them," he said. "It was a different time. It wasn't scorched-earth angry politics."

In a statement on Saturday, Mr. Carvey said that "it was an honor and a privilege to know and spend time with George H.W. Bush for over 25 years."

"When I think of those times what I remember most is how hard we would laugh," he said. "I will miss my friend."