The 27-year old Scottish attacking midfielder will make the move to Carrow Road later this week to spend the remainder of the season in the Championship with the Canaries.
"I'm glad to finally get it sorted," Dorrans told the club's official website. "I came here on the last day of the transfer window and for whatever reason it didn't work out.
"I've been in discussions ever since then, so although I was frustrated it did not get done on the final day, hopefully now I can add to a good squad.
"I spoke to the manager (Alex Neil) and a couple of the boys, and they had nothing but nice things to say about the club.
"The manager and everyone here is working to get this club back to the Premier League, and I'm excited to be here.
"With the fans right behind us, hopefully we can do that as we've got some big games coming up. I'm looking forward to getting started, and to help us push on."
Dorrans, who has been at The Hawthorns since 2008, has made 21 Premier League appearances this season for 19 starts.
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