The offensive lineman fractured his foot during a noncontact drill Tuesday. It is not known how long Smith will be out, but he will miss the Bengals' season opener on Sept. 13. Smith missed 30 days before agreeing to contract terms two days ago.
"Andre suffered a small fracture in his left foot during a non-contact drill," Bengals coach Marvin Lewis said in a statement. "Our medical staff tells me the injury will not require surgery. It's too soon to closely predict how long it will be before he can return to practice, but it looks like he's going to miss a few weeks."
Smith is currently on a roster exemption granted by commissioner Roger Goodell because of his late arrival. The exemption lasts until next week.
Smith was the sixth pick in the NFL draft and the most recent to sign. Wide receiver Michael Crabtree of the 49ers is the only unsigned first-round pick.
Sources told ESPN.com's James Walker on Tuesday that the Bengals demanded that Smith have a weight-control clause in his contract. Smith would lose a significant portion of his salary if his weight exceeded 350 pounds.
Though listed at 335 pounds, Smith tipped the scales at 364 when he showed up to sign his contract, convincing team officials to install the clause on the spot, sources told ESPN NFL Insider Adam Schefter.
Smith, whose deal includes $21 million guaranteed and maxes out at six years and $42 million, was trying to claim the starting right tackle job.
If Smith does not get his weight down to 350 pounds in the next 30 days, and he is inactive for a game, the Bengals will have the right to fine him up to 50 percent of one game check (roughly $10,000), sources told Schefter. He would have the chance to recoup the money through performance, but the incentive is there.
Since he resumed football workouts Sunday, Smith's weight already has dropped to 358 pounds, sources said.
Now, the Bengals will keep second-year lineman Anthony Collins at right tackle. Collins was a fourth-round draft pick last year and started six games at right tackle because of injuries.
The line has undergone an overhaul in the offseason. The Bengals made changes at center and at both tackle positions, hoping to upgrade an offense that finished last in the league in 2008.
The priority is protecting quarterback Carson Palmer, who was hit often last year. He broke his nose in the preseason, and tore a ligament and a tendon in his passing elbow during the regular season, forcing him to miss a dozen games.
Palmer suffered a moderate sprain in his left ankle during the first preseason game this year, and has missed the past two. He took snaps with the starting offense on Tuesday for the first time since he got hurt.
"He looked remarkably improved," Lewis said. "He looked pretty good. He was moving better."
Palmer would like to play in the final preseason game, but a source told ESPN NFL Insider Adam Schefter that Palmer will not play in that game. Palmer will play in the regular-season opener, however, the source said.
Senior writer John Clayton covers the NFL for ESPN.com. Information from The Associated Press was used in this report.