Mark Bartelstein, the agent for both players, confirmed the move on Sunday.
"This is a great fit for both guys," Bartelstein said.
Rush averaged 8.9 points and 3.6 rebounds in three seasons for the Pacers.
The Pacers considered him a part of the core group of young players, but the team eventually added enough talent at his position to make him expendable. Bartelstein said Pacers president Larry Bird and general manager David Morway worked to find an ideal setting for Rush once Paul George emerged as a starter and the team added guard George Hill in a draft-night trade.
"The Pacers have been talking about this for quite a while," Bartelstein said. "I really appreciate Larry and David for trying to find a good home for Brandon. They're loaded at the position. Brandon is a really talented guy who needs to play."
Rush showed great potential at times, but also found trouble. He was suspended for the first five games last season for violating the league's substance abuse policy.
Amundson has averaged 4.1 points and 3.7 rebounds in five seasons, primarily with Philadelphia, Phoenix and Golden State. He shoots 52 percent from the field for his career.
"The Pacers probably needed another big," Bartelstein said. "Louis is a high-energy guy that gets you extra possessions. He's a great athlete, a really good NBA player."