The 31-year-old, who has made 187 appearances in all competitions, has admitted that the opportunity to play in France was too good to turn down.
"I am aware that I don't have all that many years left in me at the top end of the game," he told Exeter's official website.
"And to get an opportunity to play in France is something I have always wanted to do at some stage in my career. Leaving the Chiefs is one of the toughest decisions I've had to make in my life.
"Exeter Rugby has had such an effect on me, not just in terms of playing, but in terms of my life as well. It's a fantastic club and I've enjoyed so many happy memories from my time here."
Clark will officially join Oyonnax at the end of the season.