Gravgaard, 30, has 18 international caps, scoring five goals.
The qualified accountant was a relatively late entrant to the professional game, joining Viborg from local side Randers in 2002.
He earned a move to Copenhagen three years later and got his first cap soon afterwards.
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Gravgaard - who can also play as a striker - hit the headlines in June 2007 when, during a Euro 2008 qualifier between Denmark and Sweden, he pulled a drunken fan away from referee Herbert Fandel during the now infamous on-pitch attack.
Nantes won promotion from Ligue 2 last season after finishing second, behind Le Havre.
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