Sigurdsson played a starring role in Iceland's stunning 2-1 win over England in the last 16 at Euro 2016, scoring his side's opening goal and being voted man of the match.
The 30-year-old centre-back has signed a two-year contract, with an option for a further year, and arrives for an undisclosed fee.
"I'm just very happy and relieved that the deal is done," Sigurdsson said in a video interview on the Fulham website.
"I've seen Fulham many times on TV, when they were in the Premier League. It's a good club and it's in London. It has everything.
"I've always wanted to play in England and I think that this is the perfect place to start and show what I can do."
Sigurdsson's Iceland team-mate Eidur Gudjohnsen, the former Chelsea and Barcelona player, tweeted: "Congrats @FulhamFC you have just signed "a beast" of a player in @sykurinn!!!"
Fulham are currently in fifth place in the Championship after four games of the season.