The Rams parted with two draft picks to acquire Barron, trading fourth- and sixth-round selections to the Buccaneers, a source told ESPN NFL Insider Adam Schefter.
The Patriots, meanwhile, gave up a fifth-round pick to the Bucs in exchange for Casillas and a sixth-round selection, a source told Schefter.
Barron, the Buccaneers' first-round draft pick in 2012, was the team's second-leading tackler this season, recording 49 tackles in seven games.
But the 25-year-old Barron reportedly struggled at times in the defensive system implemented by new coach Lovie Smith.
Barron could fill another prominent role with the Rams (2-5), who also are in last place and may need to replace starting safetyRodney McLeod, who suffered a knee injury in Sunday's loss toKansas City.
The seventh overall draft pick in 2012, Barron started all 37 games he appeared in over parts of three seasons with Tampa Bay, whichfell to 1-6 with Sunday's overtime loss to the Minnesota Vikings.
To make room for Barron on their roster, the Rams will waive quarterback Case Keenum, a source told ESPN's Adam Caplan.
The 6-foot-1, 227-pound Casillas is in his sixth NFL season, and his primary contributions come on special teams. He potentially could also add depth to New England's defense, specifically in sub packages.
The Patriots, who were already thin at the position, lost top linebacker Jerod Mayo to a season-ending torn patellar tendon Oct. 12. This is their second deal for a linebacker in a week, as they acquired Akeem Ayers from the Tennessee Titans on Oct. 21.
Casillas, who has been with the New Orleans Saints (2009-12) and Buccaneers (2013-14), has played in 55 career regular-season games (15 starts). The 27-year-old has been slowed by a hamstring injury this season, playing in five games.
To clear room for Casillas, the Patriots waived offensive lineman Chris Barker, a source told ESPN Insider Field Yates.
ESPN.com Patriots reporter Mike Reiss contributed to this report.
Barron Traded To Rams
John Clayton breaks down why he is a little bit surprised by the Buccaneers' decision to trade Mark Barron to the Rams.