The Englishman becomes only the second home-grown player in the Claymores history to sign for an NFL team.
Hart's move was concluded on Thursday evening after the player travelled to Florida at the beginning of the week for negotiations.
He will take part in the Buccaneers' training camp, which starts on Sunday, ahead of the NFL season, which kicks off in early September.
Hart admitted: "Playing in the NFL has always been my dream, so this is the biggest opportunity of my career.
"At the end of the season I moved to Bristol to work for a mate in a call centre, earning just over a fiver an hour. Now I've been handed a chance with one of the biggest sports teams in the world."
Hart, who is well-known in the NFL Europe for kicking bare foot, joined the Claymores from the London Monarchs in 1999.
Despite the Claymores' disappointing season this year, Hart enjoyed personal success, never missing a kick, hitting 10 field goals and 20 point after touchdowns.
Claymores general manager Steve Livingstone said he was delighted Hart had earned this opportunity.
"One of the aims of the Scottish Claymores, in line with all other teams in the NFL Europe League, is to produce players good enough to attract the interest of the NFL," said Livingstone.
"Rob has been a model of consistency since he joined us four seasons ago and thoroughly deserves this chance to show the Bucs what he can do."
Scott Couper, who signed for the Chicago Bears in 1997, is the only other home-grown player from the Scottish Claymores to have signed for an NFL team.