It is a deal that rewards the 25-year-old for a late-season revival that convinced the Hamburg board to activate a purchase option priced at €5m and negotiate it down by half a million.
HSV sporting director Claus Costa confirmed the deal and praised the midfielder's qualities.
"Albert quickly showed what particular added value he can give our game. With his football intelligence, his technical quality and his tactical flexibility, he brings exactly the skills that help our squad in many situations."
Grönbaek spent last season on loan at HSV from Rennes and struggled to make a consistent impression for much of the campaign before a strong run of form in the final weeks changed the club's thinking entirely. That turnaround proved sufficient to trigger a permanent move and prompt negotiations over the purchase clause.
Grönbaek has previously played in France, Denmark and England during a career that has taken him across Europe. He has also been capped at senior international level for Denmark.