The Norwegian leaves his post as Stavanger Vikings assistant coach to replace Hansen with immediate effect.
Ingebrigtsen is on a four-month trial, and was presented to the press on Monday evening at his new club. He is Rosenborg's 11th coach in the past 12 years.
"When Rosenborg calls, then you say 'yes, thanks'," said Ingebrigtsen to news website Adressa.no."This has been one of the fastest turnarounds in world history. I was on a training match with Viking football club earlier today and now I am here. I look forward to starting on the coaching field."
Rosenborg issued a club statement on Monday evening: “Kåre Ingebrigtsen is employed as the new head-coach at Rosenborg until the end of 2014. Kåre starts the position immediately. Erik Hoftun will function as assistant coach. Micke Dorsin will also have a role in the coaching team. Geir Håvard Hjelde and Jørn Jamtfall will continue and we will, in addition, use resources from the SalMar Academy.”
Kåre Ingebrigtsen is a former Rosenborg player and Norway internationalist. He also played for Lillestrøm FC and Manchester City.
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His coaching career began in 2007, with spells at Ranheim Fotball, Bodø/Glimt, and then as assistant coach for Vikings.
Ingebrigtsen's first match in charge of Rosenborg will be in the Europa League against Sligo Rovers on Thursday. Then three days later his side play league leaders Molde.