The Iceland international joined the Swans on loan in January, after struggling to nail down a first-team berth in Germany and proved to be a huge success for Brendan Rodgers' men, scoring seven goals in 19 appearances as the south Wales side secured an 11th-placed Premier League finish.
Swansea have moved quickly to ensure that deal became permanent, making the 22-year-old their first signing of the summer after also agreeing personal terms.
Sigurdsson began his senior career at Reading, where Rodgers previously spent an ill-fated spell in charge, making 38 league appearances in his first season but just four in his second.
The Icelandic attacker has seven caps for his national side and scored his first international goal during a Euro 2012 qualifier against Portugal in 2011.
His transfer fee almost doubles the club record of £3.5m paid to Watford to secure the services of striker Danny Graham in July 2011.