Coples, 25, played in college at North Carolina and was the 16th overall pick in the 2012 draft. He played three-plus seasons for the Jets but was released last November.
The Miami Dolphins signed the 6-foot, 6-inch, 289-pound Coples but released the reserve in February to clear salary-cap space. He was reportedly due to earn $7.5 million.
Coples has 16½ career sacks.
The Rams have executed four transactions involving defensive linemen since the end of last season.
Hayes and Sims re-signed with the Rams, and Coples now joins a line that also features ends Robert Quinn and Matt Longacre and tackles Aaron Donald, Michael Brockers, Louis Trinca-Pasat, Ethan Westbrooks and Doug Worthington.
Tackle Nick Fairley, who played last season for the Rams, is an unrestricted free agent.