The Norwegian international has put pen-to-paper on a deal that sees him contracted to the club until the summer of 2015, with the option of a further year. Arriving from Manchester United for an undisclosed fee.
"I've enjoyed my time here, we've a great group of lads and I think I've settled in well," said King.
"I'm happy to be here and I'm really looking forward to the next two-and-a-half years that I've signed for.
"Everybody has made me so welcome here, I feel comfortable here, I'm happy to make the deal permanent."
Signed initially on loan at the end of November, the 20-year-old attacker has proved to be a popular addition to the Rovers squad, scoring two goals in his eight outings to date.