The Rockies also sent the Reds $1.9 million to make up the difference in salaries.
The 35-year-old Bichette hit 34 homers and drove in 133 runs for the Rockies last season. The Reds are counting on him to replace Vaughn, who hit 45 homers and drove in 118 runs but is unaffordable as a free agent.
Hammonds was a backup outfielder last season and Belinda was a middle reliever after coming back from a diagnosis of multiple sclerosis.
Bichette, one of the original Rockies, was the franchise's career leader in hits and runs.
"Our vision for this team is a more athletic, more versatile group of contributors," Rockies general manager Dan O'Dowd said. "This is the first step in that direction. It also gives us important payroll flexibility."
The trade was the latest major change for the franchise, which had general manager Bob Gebhard retire last August and manager Jim Leyland retire following a 72-90 season, the second-worst in franchise history.
"I gave them everything I could," Bichette said. "We part even and I still remain in their hearts and them in mine. It's emotional. I'll drop a few tears, I'm sure, because it was the bulk of my career and this is what I love to do."
The trade wasn't a surprise. Bichette mentioned to the Rockies early last season that he missed his family, which lives in Orlando, Fla., and would welcome a trade to an East Coast team that had spring training in Florida.
The Reds train in Sarasota, Fla., and narrowly missed out on the playoffs last season, satisfying Bichette. He waived his right to block the trade.
"The family situation was the most important," he said.
The Reds are coming off a 96-win season for which Vaughn was largely responsible both as a clubhouse leader and in that he carried the team in September.
The Reds missed out on the playoffs by losing a one-game tiebreaker for the NL wild card to the New York Mets and were determined to get another proven cleanup hitter. Vaughn filed for free agency last Thursday.
General manager Jim Bowden thinks Bichette, who hits for higher average and strikes out less than Vaughn, can fill his role.
"He's a notch below Greg defensively, but he's adequate," Bowden said. He's a better hitter than Greg Vaughn. Greg Vaughn has a little more power. They both produce runs."
The Reds began talking to the Rockies about Bichette during the season and at one point were working on a five-player deal, Bowden said. The last snag was the money: The Reds had a $35 million payroll last season and aren't expected to increase it substantially.
That left Vaughn out of the picture. Initial talks with his agent indicated he was looking for a huge pay increase.
"He was looking for four or five years in the $10 million to $12 million range (per year)," Bowden said. "We're not able to do that financially."
The trade with Colorado was finished when the Rockies agreed to give the Reds $1.9 million to offset his salary.
Bichette, who turns 36 on Nov. 18, is entering the second year of a three-year deal worth $21 million. He'll make $6.5 million in base salary each of the next two seasons and could make as much as $250,000 per year in performance bonuses.
Hammonds, 28, hit .279 with 17 homers and 41 RBIs in 123 games last season. He was the least-used member of a crowded outfield. He'll make $3.1 million in base salary next year, the last on his contract.
The 33-year-old Belinda lost his job as setup man after sitting out the end of last season after being diagnosed with multiple sclerosis. Biceps tendinitis forced him onto the disabled list as the start of this season, and he went 3-1 with a 5.27 ERA and two saves in 29 relief appearances.
The Reds exercised their option and kept Belinda under contract for next season at $1.5 million.
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