Nantes confirmed on Thursday that Sigthorsson, 25, had signed a five-year contract.
He has spent the last eight years in the Netherlands, first with AZ Alkmaar and then with Ajax, with whom he won three successive league titles from 2012-2014.
He has scored 17 goals in 29 games for Iceland, including the winner in their 2-1 victory over the Czech Republic in Euro 2016 qualifying last month.
Two points clear at the top of Euro 2016 qualifying Group A with four games left, Iceland are on course to qualify for a major tournament for the first time.