Kubina, a 2004 All-Star, has one year left on his deal at $5 million.
Kubina had become injury-prone in recent years, but played a full 82 games in 2008-09 for the first time since the 2001-02 season. He bolsters a young defensive corps in Atlanta that boasts 2008 third overall selection Zach Bogosian and 24-year old Tobias Enstrom.
Thrashers general manager Don Waddell said the 32-year-old Kubina will give Atlanta one of its strongest defensive lines.
"I feel it's one of the best groups of top four defensemen we've probably ever had here in the history of the Thrashers," Waddell said, referring to Kubina, Bogosian, Enstrom and Ron Hainsey.
Kubina was part of the "Frozen Five" who refused to waive their no-trade clauses at the 2008 trade deadline. Out of the five, only Tomas Kaberle remains with Toronto. Mats Sundin, Bryan McCabe and Darcy Tucker parted ways with Toronto prior to the 2009 season.
Leafs general manager Brian Burke had been looking to move Kubina since he took over from interim general manager Cliff Fletcher in November. Burke is in the process of retooling the Leafs into a eam similar to the one he built as general manager of the Anaheim Ducks that won the Stanley Cup in 2007. Burke added physical defenseman Mike Komisarek and enforcer Colton Orr earlier Wednesday.
Exelby is a player with size and grit who will deepen Toronto's backline. He is known mostly for his defensive and physical capabilities and has 46 points and 511 penalty minutes in 357 career games.
Waddell said it was painful to trade Exelby, a 1999 Atlanta draft pick who appeared in more games than any defenseman in team history.
"It's disappointing to have to trade one of your favorite players," said Waddell, who said Exelby "did everything we ever asked of him."
Stuart had five goals and three assists in 33 games.
"Stuart started coming into his own last year and was a good player for us, but when you make a deal and when you get a good player back, you have to give something up," Waddell said.
Stapleton scored one goal in four games with the Maple Leafs last season.
Kubina matched his career high with 40 points for Toronto last season. He had 14 goals and 26 assists.
Information from ESPN.com's Pierre LeBrun and Scott Burnside and The Associated Press contributed to this report.