Selected sixth overall, Claiborne inked a four-year deal worth $16.26 million, according to sources.
The cornerback will be able to practice Wednesday at Valley Ranch when pre-training camp workouts begin.
The monetary value of the deal, which includes a signing bonus of $10.27 million, had been in place for weeks, but the offset language had been a sticking point for almost all of the top picks in the first round. According to sources, the deal contains no offset language.
ESPN business analyst Andrew Brandt explained the offset issue in a recent "Football Today" podcast.
"Teams want language in the contract, these are fully guaranteed contracts, saying that if they cut the player at some point and he signs with another team, they offset their guarantee," Brandt said. "So they cut a player, he signs for a million dollars somewhere, that million comes off what they owe."
On draft night, the Cowboys made the bold move to get Claiborne by sending their first- (No. 14 overall) and second-round picks to St. Louis to land the top-rated defensive player on their draft board. However, Claiborne was unable to practice all offseason while recovering from surgery to repair ligaments in his wrist.
Cleared medically, Claiborne will practice Wednesday when rookies, first-year players, quarterbacks and selective veterans report for camp. The first full-squad workout in Oxnard, Calif., takes place July 30.
In 33 games at LSU, Claiborne intercepted 11 passes and knocked down 12 others. Six of his interceptions came in 2011, when he was named a first-team All-American and winner of the Jim Thorpe Award, which is given to college football's best defensive back.
The Cowboys have revamped their secondary in 2012 by drafting Claiborne and signing cornerback Brandon Carr to a $50 million deal as a free agent to go with veterans Mike Jenkins and Orlando Scandrick.
The Cowboys are counting on Claiborne to be ready to play as soon as possible, especially with Jenkins likely to start training camp on the physically unable to perform list because of offseason shoulder surgery.
The last time Dallas drafted a Day 1 starter at cornerback was Terence Newman, the No. 5 overall pick in 2003.
Calvin Watkins of ESPNDallas.com contributed to this report.