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Meacham informed TODAY's Willie Geist that he read the speech to Bush before the former president's death on Friday at 94.
He received a response from the 41st president that was typical of the humor and humility frequently cited in remembrances of Bush at the memorial in Washington, D.C., on Wednesday.
"That's a lot about me, Jon,'' Bush replied after hearing the eulogy.
While hearing your own eulogy while you're still alive may sound unorthodox, the Pulitzer Prize-winning historian and the former president had a lasting bond stretching back 20 years, according to The New York Times.
Meacham became Bush's biographer, resulting in the 2015 best-seller "Destiny and Power: The American Odyssey of George Herbert Walker Bush."
Bush provided Meacham with access to hundreds of hours of audio diaries and allowed him to extensively interview him along with family members and close friends for the book, forging a close connection between the two.
Meacham's well-received eulogy connected Bush's life of public service in politics to his experiences in World War II, where he survived being shot down over the Pacific as a Navy pilot in 1944.
"Through the ensuing decades, President Bush would frequently ask, nearly daily, he'd ask himself, 'Why me? Why was I spared?''' Meacham said. "And in a sense, the rest of his life was a perennial effort to prove himself worthy of his salvation on that distant morning."
Meacham also spoke glowingly about Bush's character.
"If you were down, he would rush to lift you up,'' he said. "And if you were soaring, he would rush to savor your success. Strong and gracious, comforting and charming, loving and loyal, he was our shield in danger's hour.
"... His life code, as he said, was, 'Tell the truth. Don't blame people. Be strong. Do your best. Try hard. Forgive. Stay the course.' And that was and is the most American of creeds."
Meacham's remembrance was followed by ones from former Canadian Prime Minister Brian Mulroney, former Sen. Alan Simpson of Wyoming, and an emotional eulogy by Bush's son, former President George W. Bush.
The service in Washington will be followed by one on Thursday in Houston, where Bush lived, before he is laid to rest on the grounds of his presidential library at Texas A&M University.
Meacham's eulogy ended by returning to the theme of Bush's military service informing the rest of his life.
"The workings of providence are mysterious, but this much is clear: that George Herbert Walker Bush, who survived that fiery fall into the waters of the Pacific three-quarters of a century ago, made our lives and the lives of nations freer, better, warmer and nobler.
"That was his mission. That was his heartbeat. And if we listen closely enough, we can hear that heartbeat even now. For it's the heartbeat of a lion, a lion who not only led us, but who loved us."