The Jets receive a sixth-round pick in the 2012 draft from the Colts, a league source said. They send Stanton and a 2012 seventh-round selection to the Colts -- the pick they acquired in the Tim Tebow trade.
The Jets signed Stanton last Friday to be their No. 2 quarterback, but their plans changed when they completed the trade for Tebow.
Stanton became upset and asked the Jets to trade or release him.
Stanton's agent, Deryk Gilmore, chided the Jets via Twitter not long after the trade was announced.
"Excited for the #Colts to get @drewstanton a leader, winner, and QB not a gimmick to sell tickets," a post on Gilmore's Twitter page said.
But a few minutes later, the tweet was deleted, and a new post struck a conciliatory tone.
"Jets are a class organization for doing the right thing," Gilmore's tweet said. "I believe in my players and know what they can do."
Stanton signed a one-year contract for $1.25 million, including a $500,000 signing bonus the Jets have paid, a source said.
"It's a big relief to have this over," Stanton told The Associated Press. "I'm excited about the opportunity in Indy. I can't wait to get there and get down to business. ... A big reason I signed with the Jets was the opportunity they gave me. Circumstances changed, of course, but the Jets were nothing but professional and helped me get to Indy."
When the Jets traded for Tebow on Wednesday night, Stanton's agents presented them with a list of teams they preferred. The list consisted of the Colts, Denver Broncos, Pittsburgh Steelers, Green Bay Packers and St. Louis Rams, according to a source.
Stanton said he was promised the No. 2 job by the Jets, and he became upset when general manager Mike Tannenbaum publicly declared Tebow the No. 2 quarterback.
"Drew is a big, athletic and strong-armed quarterback that has the intangible package we are looking for at the position," said Colts general manager Ryan Grigson. "He is a young, but seasoned quarterback that has won games in this league and we look forward to seeing him contribute to this team and our collective vision for the future."
Information from The Associated Press was used in this report.