The Predators also receive a conditional 2013 draft pick, in exchange for forwards Blake Geoffrion and Robert Slaney, plus a 2012 second-round pick.
Gill, 36, brings defensive-zone responsibility and a wealth of playoff experience to a Predators group that advanced past the first round for the first time last season.
"Hal Gill brings our team a number of elements -- size, penalty-killing ability, depth on defense and playoff experience," said Predators General Manager David Poile. "He has won a Stanley Cup and played a combined 69 playoff games over the last four years, experience that significantly benefits our team."
In 14 seasons, Gill has 36 goals and 143 assists in 1,047 regular-season games with the Bruins, Maple Leafs, Penguins and Canadiens, and also has 105 games of playoff experience, including back-to-back Stanley Cup Final appearances with the Pittsburgh Penguins in 2008 and 2009. He was part of the Pens' top shut-down pairing in 2009 when the Penguins won the Stanley Cup.
Gill can become an unrestricted free agent if not signed by July 1.
"First and foremost, I would like to thank Hal Gill for his contribution to our team over the last three seasons," said Canadiens GM Pierre Gauthier. "His leadership and commitment were much appreciated. I wish him the best of luck with his new team. We are also very pleased to have acquired a second-round draft selection in 2012 and two young prospects for our organization. We look forward to working together with Blake Geoffrion and Robert Slaney to continue their progression."
Gill said he wasn't surprised at being dealt.
"I was waiting for it to happen," he said during an interview after the first period. "It's always a surprise when it does happen, but I'm excited and I think this is a good opportunity for me and hopefully I can help the team out.
"I know a few guys [on the Predators]. I played with Sergei Kostitsyn in Montreal and I know guys from the U.S. program and I work out with a couple guys, so (there are) some familiar faces and some new faces as well."
Geoffrion and Slaney have been playing with the Predators' American Hockey League team, the Milwaukee Admirals.
In 22 NHL games this season, Geoffrion has three assists, but hasn't played in the League since Dec. 26, 2011. In 20 AHL games, the 6-foot-1, 190-pound left wing has two goals and seven assists.
On his personal Twitter account, Geoffrion posted: "Want to thank all of the fans of nashville for all the support over the last two years.....going to miss you guys...."
Later he added: "Wow..what a day...its bitter sweet, going to miss my hometown of nashville, but going to where it all started with my family...MONTREAL!!!"
If Geoffrion plays 40 NHL games next season, the Predators will get a 2013 fifth-round pick from the Canadiens.
A 2006 second-round pick of the Predators, Geoffrion is the grandson of Hockey Hall of Famer Bernie "Boom Boom" Geoffrion, and the great-grandson of Hockey Hall of Famer Howie Morenz. He also was the first player raised in Tennessee to play for the Predators when he made his NHL debut last season.
Slaney, a 6-2, 203-pound left wing, has split the season between the AHL and the ECHL Cincinnati Cyclones. He had one assist in nine AHL games, and eight goals and five assists in 23 ECHL games.
Contact Adam Kimelman at firstname.lastname@example.org. Follow him on Twitter: @NHLAdamK
I don't have a crystal ball. Predicting is a real complicated thing. If we stay healthy, have enough depth and get the good goaltending we think we're going to have, you can go all the way. But a lot of things have to happen. There's going to be a lot of teams that think the same thing. Everyone made deals. We're all are optimistic about where we'll end up.
— Rangers general manager Glen Sather after being asked if he's constructed a team that can win the Stanley Cup before their 4-1 win against the Predators on Monday