Muller joins Hamburg from Mainz
Muller becomes the latest arrival in what has been an impressive transfer window for Hamburg, who have already added extensively to their squad ahead of the new Bundesliga season.
The 26-year-old follows the likes of striker Pierre Michel-Lasogga and defender Johan Djourou, both of whom made loan deals permanent, and midfielder Valon Behrami to the Imtech Arena.
His move to the six-time German champions comes after an impressive campaign with Mainz in 2013-14, in which he scored 10 goals in all competitions.
Mainz are competing in the Europa League this season following a seventh-place finish last term.
However, Muller has rejected the opportunity to play European football to instead help one of Germany's historic football powers bounce back after narrowly avoiding relegation in a play-off with Greuther Furth.
"I wanted to take the next step in my career," Muller told Hamburg's official website. "Going to a big club in such a city.
"I have had very good discussions with those responsible and wanted to do something new again.
"I am now 26 years old and want to develop myself further here."