The 28-year-old winger, who had been without a club since leaving Pittodrie this summer, has penned a two-year deal.
A move to Mo Johnston's Toronto FC fell through but Smith is now in line for his debut in Saturday's game against the Columbus Crew.
Rapids manager Gary Smith said: "I think we are looking to the future with Jamie Smith.
"Jamie has been brought in predominantly as a wide player. He is versatile enough to play on both flanks as well as an attacking midfielder player."