The agreement guaranteed that the USC quarterback would be the No. 1 selection. Terms were not disclosed.
Last year's top pick, quarterback David Carr, agreed to a seven-year contract with Houston Texans prior to the draft. That deal has a base value of $21.5 million but could be worth $58 million if every incentive is reached.
"We are very excited to both select and sign Carson," Bengals Coach Marvin Lewis said. "We're pleased to be able to have him start learning our system immediately. I believe strongly he has a great NFL future. I'd like to congratulate Carson and his family, as well as our negotiating team in concluding the deal."
The Detroit Lions, who own the second pick in the draft, are now free to finalize a contract with a prospective selection once the Bengals' deal is approved by the NFL.
Palmer will be the first Heisman Trophy winner to be selected No. 1 since quarterback Vinny Testaverde went to the Tampa Bay Buccaneers in 1987.
Palmer becomes the second Heisman Trophy winner to be drafted by the Bengals. The first was running back Archie Griffin of Ohio State, a two-time Heisman winner who was the 24th overall pick in 1976.
The last player from Southern California to go first overall was receiver Keyshawn Johnson in 1996.
Lewis already has named Jon Kitna as his starter for 2003, but the future of the struggling franchise ultimately will rest with Palmer. The Bengals selected Akili Smith with the third overall pick in 1999, but Smith has been a bust, prompting the Bengals to look elsewhere.
Cincinnati has not made the playoffs since 1990.
The 23-year-old Palmer threw 72 touchdown passes and has 49 interceptions in 45 games at USC. He completed 288 of 458 passes for 3,639 yards and 32 touchdowns with 10 interceptions in his breakout senior season.
Palmer capped the year by throwing for 303 yards and a touchdown in an Orange Bowl victory over Iowa.