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Hegerberg, 26, joins the club after spending the past two seasons at French side Paris Saint-Germain.
"Finally I can tell you all that this season I have chosen to play for Roma," Hegerberg said.
"I cannot wait to meet you all and get started."
Hegerberg becomes the club's fifth addition of the summer - following this week's announcements about the signings of Kaja Erzen, Andressa Alves, Amalie Thestrup and Manuela Giugliano.
Like those four, Hegerberg is a full international - having represented Norway at senior level on 30 occasions.
She is the sister of Olympique Lyon striker Ada, who this year became the first-ever winner of the Women's Ballon d'Or.
Andrine spent the last two seasons with PSG, winning the Coupe de France in 2008.
She has previously played for Birmingham City, Gothenburg, Turbine Potsdam, Stabaek and Kolbotn.
Last season Betty Bavagnoli's Giallorosse finished fourth in Serie A - in what was the club's first-ever professional campaign.
Welcome to Roma, Andrine!
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