Rhodes, 41, was 3-3 with a 4.81 ERA in 32 relief appearances and was on the club's Opening Day roster. Texas has 10 days to trade, release or outright Rhodes to the minor leagues.
The move was necessary after the Rangers traded Double-A pitching prospects Joe Wieland and Robbie Erlin to San Diego on Sunday, a few hours before the trade deadline, for the right-handed Adams. The 33-year-old Adams is 3-1 with a 1.13 ERA in 48 relief appearances, all with the Padres in 2011. He has 49 strikeouts and nine walks and opponents are batting just .155 against him.
The Rangers acquired Rhodes in the offseason after he had an All-Star 2010 season for the Cincinnati Reds. He had a 2.29 ERA in 55 innings pitched and held opponents to a .196 average, with right-handers hitting .182 against him.
But Rhodes hasn't been able to carry over that success to this season.
Opponents have batted .289 against Rhodes, who has had particular issues getting righties out (.333 average). Because Adams and newly acquired right-handed reliever Koji Uehara have performed well against lefties, the club did not need Rhodes as a purely situational lefty.
"We didn't feel like we needed two left-handers" in the bullpen, manager Ron Washington said. "It's a very tough decision to move Arthur.
"We picked up two pretty good pieces to put in that bullpen, and someone had to be the one to move."
The AL West-leading Rangers last winter signed Rhodes to a one-year contract that guaranteed him $1.4 million, plus incentives. There was also a $4 million club option for 2012.
Richard Durrett covers the Rangers for ESPNDallas.com. Information from The Associated Press was used in this report.