In return, the Patriots have traded tight end Michael Hoomanawanui to the Saints.
The NFL Network and Boston Globe earlier reported news of the deal.
Hicks, who is 6-foot-5, 324 pounds, is a fourth-year player out of Regina (Canada). He has appeared in 47 career regular-season games, with 33 starts. He started the first three games for the Saints this season.
The move is a bit surprising on the Saints' end, because Hicks, 25, has been a starter for the past three years and showed so much potential early in his career. He also appeared to have a strong training camp this summer after he said he shed 20 pounds.
But the breakout season that many have expected over the past two years hasn't materialized. Hicks was demoted to a part-time role during Week 2 and has only two tackles this season with zero sacks.
His departure is the latest big move for a Saints team that has turned over nearly half the roster in 2015.
Though the Hicks part of the trade is a surprise, coach Sean Payton tipped the Saints' hand earlier Wednesday when he mentioned the teams' need for a third tight end to go along with veterans Benjamin Watson and Josh Hill.
"We're still short at tight end, and that's an area that concerns me," Payton said. "Typically, we've always had three with one fullback. ... That's an area we've looked at and will continue to do so."
Hoomanawanui, a six-year veteran, has been No. 4 on the tight end depth chart this year. He played 22 offensive snaps over the first two games of the season, but was a healthy scratch in Sunday's win over the Jaguars.
Hoomanawanui entered the NFL as a fifth-round draft choice of the St. Louis Rams in 2010. He was released by the Rams in September 2012 and signed by the Patriots three days later.
He was a durable player for New England from 2012 to 2014, playing in 50 games with 32 starts, including the playoffs. He totaled 29 receptions for 381 yards over that span.
Hoomanawanui slid down the depth chart in 2015 after the Patriots signed former Bills tight end Scott Chandler as a free agent and then traded for former Detroit Lions offensive tackle/tight end Michael Williams. Chandler and Williams have had larger roles than Hoomanawanui on game day.
At 6-foot-4 and 265 pounds, Hoomanawanui can play on the line of scrimmage as an in-line blocker, but was also flexed out at times.
Hicks and Hoomanawanui have contracts that expire after the 2015 season.
The two teams held joint practices in mid-August before their preseason game, giving both clubs some extra practice tape to analyze of both players.
Information from ESPN.com Saints reporter Mike Triplett contributed to this report.