Soto, 29, has been in the Cubs system since he was drafted in 2001. The trade was reported Monday night as Soto was pulled from the Cubs' game against the Pirates during the sixth inning, but wasn't finalized until Tuesday morning.
As he left the game Monday, Soto was greeted in the dugout by handshakes and hugs from his teammates. He was batting .199 in 52 games. He missed time because of a torn meniscus in his left knee.
The 2008 National League Rookie of the Year, Soto will be paid $4.3 million this year, and he was on track to go to arbitration for the third time this offseason.
"When you get a good feel with a catcher, that's hard to replace," Cubs pitcher Jeff Samardzija said. "We'll make sure we get [backup Steve Clevenger] back there to catch all my flat grounds and my bullpens."
The defending American League-champion Rangers lead the AL West division by 3 1/2 games.
The Cubs were expected to call up catcher Welington Castillo from Triple-A Iowa with Clevenger taking over as the regular catcher. Clevenger was in the weight room during Monday's game, preparing in case he was needed to pinch-hit when he got word that he had to get into the game to catch.
"It was a little strange, a little weird," Clevenger said. "Geo's always been good to me. I've been with the Cubs since '06, and in big league camp since '08, and he always treated me really well. I'm going to miss him. I learned a lot for him and he treated me well."
Brigham, 24, was 5-5 with a 4.28 ERA in 21 starts for Double-A Frisco. He went 3-1 with a 3.60 ERA (16 ER/40.0 IP) in 21 relief outings last season. He was named to this year's mid-season Texas League All-Star team.
Brigham was selected in the sixth round of the 2006 First-Year Player Draft out of Central Florida Christian Academy in Orlando, Fla. He is 27-42 with one save and a 4.40 ERA in 135 Minor League appearances.
Carrie Muskat is a reporter for MLB.com. She writes a blog, Muskat Ramblings, and you can follow her on Twitter @CarrieMuskat. This story was not subject to the approval of Major League Baseball or its clubs.