Borbon, who was hitting .270, left Friday's game against the Los Angeles Angels when running down a fly ball in the top of the seventh inning.
To replace Borbon, the Rangers called up Endy Chavez from Triple-A Round Rock. The team also moved pitcher Brandon Webb (rotator cuff) from the 15-day disabled list to the 60-day DL. Webb is eligible to come off the DL on May 30.
Chavez is in the starting lineup for Saturday's game and leading off.
"It's a great feeling," said Chavez, who came back from a torn ACL and MCL and underwent microfracture surgery in 2009 while playing for the Seattle Mariners. "I was working for almost two years and it's good to be back and now I have an opportunity to play over here."
The Rangers now have their opening day outfield on the disabled list.
Manager Ron Washington said he's never seen that happen so soon into a season.
But there is hope.
Left fielder and reigning AL MVP Josh Hamilton took batting practice for the second consecutive day and hopes to continue that next week when the club goes to Chicago for a two-game road trip.
Hamilton, on the DL since April 13 with a fractured bone in his right shoulder, has missed the last 28 games, which includes Saturday's game against the Angels.
After taking more BP in Chicago, Hamilton will undergo a CT scan to check on the healing process. If all goes well with the tests, Hamilton will do a rehab stint where Washington said he hopes to get him about 20 at-bats.
"The biggest thing is not falling on it, just being careful with it," Hamilton said. "That's going to be the biggest problem for a little while at least until the bone is starting to heal. That's one reason I will DH a little bit."
Starting right fielder Nelson Cruz, on the DL since May 7 with a strained right quad muscle, ran the bases for the first time on Saturday.
Washington would like to also get Cruz some rehab at-bats, about 20, before bringing him to the big club. If Cruz feels good physically and does well with his rehab, Washington said the right fielder might play next weekend in Philadelphia.
"There is light at the end of the tunnel," Washington said. "In a couple of weeks, we'll have some big guns back."
Calvin Watkins covers the Rangers for ESPNDallas.com.