Demouge joined Bournemouth from Utrecht 12 months ago, but he made only two appearances during his time with the club.
He linked up with Roda JC on loan in the January transfer window and had his contract with Bournemouth cancelled on Friday.
The 31-year-old impressed during his loan spell at Roda JC, and the Dutch club's managing director Marcel van den Bunder is happy to have snapped up Demouge on a permanent basis.
"Frank has shown in the last six months that his mentality and style of play fits very well into our club," Van den Bunder told Roda JC's official website.
"It is not surprising that, after he has been able to terminate his contract at Bournemouth, he has quickly reached an agreement with us."