The swing quartet, who reached the finals of the ITV talent show earlier this year, are set to release their debut album in 2015.
Jack Pack wowed crowds during the competition with their versions of big band classics including Frank Sinatra's 'That's Life' and Nina Simone's 'Feeling Good'.
"It's a dream come true for us to be signing to Syco Music," said Jack Pack. "We'd like to thank Simon Cowell for believing in us and giving us this opportunity.
"We can't wait to get to work in the studio and try to repay our supporters for the faith they have shown in us."
Meanwhile, music mogul and X Factor boss Simon said: "I love this era of music, it only works if you've got great voices and if you respect the time and Jack Pack win on both counts. I've always wanted to work on this type of record so I am looking forward to us working together."
Jack Pack are made up of Sean Ryder Wolf, Alfie Palmer, Andrew Bourn and new member Martin McCafferty. Martin replaces original member Adam Chandler, who left the group in June.
Their debut album is expected to take influences from the Rat Pack and swing music styles.
Jack Pack aren't the only act from this year's BGT to sign record deals. The show's winners Collabro signed a deal in June, while young duo Bars and Melody announced their deal a week later. All acts have been signed to Simon's Syco Music.
Elsewhere, the show's runner-up Lucy Kay signed a multi-album deal with Sony Classical in June.
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Watch Jack Pack on BGT below: