Cousins had been in conversations with the Nuggets for two weeks and will be eligible for six games in that stretch, beginning with Friday's home game against the Memphis Grizzlies.
Cousins, 31, averaged 9.1 points and 5.8 rebounds in 17 games for the Milwaukee Bucks this season. He had 15 points, 10 rebounds and 3 steals in his final game on Jan. 6 before Milwaukee decided against guaranteeing his contract for the rest of the season.
"We wouldn't have been able to get through this difficult stretch of the season as successfully as we did without DeMarcus," Bucks general manager Jon Horst told ESPN at the time. "At the end of the day, we made a strategic decision to have an open roster spot, but there's nothing that would prevent us from partnering with DeMarcus again down the road. He was so good for us, and hopefully we helped him, too."