"After passing his medical on Wednesday... Danish striker Kasper Dolberg completed his transfer to OGC Nice on Thursday morning," Nice said in a statement.
Ajax said that 21-year-old Dolberg leaves the Dutch champions for a fee of 20.5 million euros ($22.7m), but neither club revealed the details of the attacker's contract, believed to be long-term.
The Danish international striker was in the stands at Nice on Wednesday to see his new team lose 2-1 to Marseille in a match suspended for homophobic abuse from the stands.
The match was the first of the Ineos era following Monday's takeover, and Bob Ratcliffe, the brother of billionaire Jim Ratcliffe who owns the petrochemical giant, was in the stands to see the team taken off the pitch for over 10 minutes following offensive chants and banners.
Dolberg made 39 appearances for Ajax last season, scoring 12 times as the Amsterdam club won a domestic double and reached the Champions League semi-finals.
He played 119 times in all competitions in his three seasons with Ajax, scoring 45 goals.