Frost, celebrated for his 1977 talks with Richard Nixon that extracted an unexpected apology from the disgraced US president over the Watergate scandal, died Saturday.
Operator Cunard said the ship left its British home port of Southampton on Saturday on a 10-day Mediterranean.
"Sir David Frost died of a heart attack last night aboard the Queen Elizabeth where he was giving a speech," his family said in a statement.
"His family are devastated and ask for privacy at this difficult time," the statement said. "A family funeral will be held in the near future and details of a memorial service will be announced in due course."
Frost's interviewees read like a who's who of the rich and famous, from big names in show business to world leaders, including South African anti-apartheid icon Nelson Mandela.
Frost was the only person to have interviewed the last eight British prime ministers and the last seven US presidents before Barack Obama, and the last person to have interviewed the last shah of Iran, the Mohammed Reza Pahlavi.
Other subjects included Mikhail Gorbachev, Vladimir Putin, Yasser Arafat, F. W. de Klerk, Jacques Chirac and Benazir Bhutto.
"Hello, good evening and welcome" became his catchphrase, starting off interviews with a friendly veneer that belied a blunt determination to extract information.
"His scrupulous and disarming politeness hid a mind like a vice," said Menzies Campbell, former leader of Britain's Liberal Democrats. "David Frost could do you over without you realising it until it was too late."
The lengthy interviews with Nixon were crucial for both men -- Nixon was hoping to salvage his reputation for history, while Frost wanted to add another feather to his cap of famous interviews .
In the end, Frost wrung a mea culpa from Nixon over Watergate, the dirty tricks scandal which prompted his resignation in 1974 and left a lasting scar on the US political landscape.
"I let down my friends, I let down the country," the former president said.
Frost told BBC television in 2009: "We knew what we were trying to do ... and in the end his 'mea culpa' went further than even we had hoped.
"At the end of that I think we were aware that something sort of historic had happened and we'd gone further than expected."
The encounter was turned into a play entitled "Frost/Nixon", which was adapted into a 2008 film with Michael Sheen playing Frost and Frank Langella as Nixon. It was nominated for five Oscars.
Outside world affairs, Front's roster included Orson Welles, Tennessee Williams, Noel Coward, Elton John, Woody Allen, Muhammad Ali, the Beatles, Clint Eastwood, Anthony Hopkins, John Gielgud, Norman Mailer, Warren Beatty among countless others.
British Prime Minister David Cameron hailed Frost as "an extraordinary man -- with charm, wit, talent, intelligence and warmth in equal measure.
"He made a huge impact on television and politics. The Nixon interviews were among the great broadcast moments -- but there were many other brilliant interviews," Cameron said in a statement.
"He could be -- and certainly was with me -- both a friend and a fearsome interviewer."
The son of a Methodist minister, David Paradine Frost was born in Kent, southeast England, on April 7, 1939.
Fresh out of Cambridge University, he presented the BBC's groundbreaking "That Was The Week That Was", which took an unprecedented satirical look at the week's news between 1962 and 1963.
A globetrotter, Frost revelled in the Concorde jet-set high life, presenting five programmes a week in the United States and three in Britain.
In 1983, he married Lady Carina Fitzalan-Howard, second daughter of the Duke of Norfolk -- the premier duke in the English nobility. They had three sons.
A successful businessman, Frost was knighted in 1993, becoming Sir David.
The broadcaster wrote 17 books, produced several films and started two British television networks, London Weekend Television and TV-am.
He began working for Al Jazeera in 2006.
