Guard Kristy Wallace heads to Indiana in the deal.
A three-time WNBA All-Star and three-time WNBA All-Defensive team selection, Robinson was the No. 6 pick in the 2011 draft out of Oklahoma, where she helped lead the Sooners to two Final Four appearances.
She turns 34 in May, when the WNBA's 27th season begins.
Robinson began her career in San Antonio, where she played five seasons. She also played for the Phoenix Mercury, Minnesota Lynx and Las Vegas Aces, the franchise that was formerly in San Antonio before moving in 2018.
She spent the past two years with the Fever, averaging 7.4 points in 2022 and 9.9 in 2021.
"Danielle's skillset, competitiveness, and basketball IQ will prove her to be a great addition to our team," Dream general manager Dan Padover said in a statement. "We know she will continue to bring the same energy to the floor that she has brought throughout her entire professional career.
"Danielle is the exact type of professional we want to bring into this organization. Her leadership and experience will be a huge asset for our young talent as they continue to grow into the future of the Dream."
The Dream finished 14-22 last season and saw No. 1 pick Rhyne Howard earn Rookie of the Year honors. They are hoping to supplement the talent around Howard via free agency (players can begin signing Feb. 1) and the 2023 draft (Atlanta has the Nos. 3 and 8 picks in the first round).
Wallace was a second-round draft pick (No. 16 overall) in 2018 after her senior season at Baylor was cut short by a knee injury. She didn't get on a WNBA roster until last season with the Dream. In the meantime, she played professionally in her native Australia.
Wallace, who turned 27 on Jan. 3, averaged 6.6 points, 2.3 rebounds and 2.2 assists last year for Atlanta.