The club revealed earlier this month that Costa had performed a U-turn on her decision to take at the French outfit, citing personal reasons for her departure.
The 36-year-old hit the headlines in May when she was initially handed the job, giving her the opportunity to coach at a higher level than any other woman in European football.
However, that honour will now fall to former France captain Diacre, who signs a two-year contract with the club.
A statement on Clermont's official website on Saturday read: "Corinne Diacre has been appointed coach of Clermont Foot. She is committed for the next two seasons and will take office early next week."
Diacre made 121 appearances for France and served as assistant coach for the women's national team from 2007 to 2013.
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