Ablyazov is accused by Kazakhstan, Russia and Ukraine of embezzling billions of dollars during his time as chairman of BTA Bank.
A former trade minister, he fled Kazakhstan after the bank was taken over by the government in 2009, and officials later found huge gaps in its accounts.
He was arrested on the French Rivieria in July 2013.
The case has gone back and forth in France -- with the extradition approved in January 2014, overturned by an appeals court and then reconfirmed in October -- but the government still had to give its final approval.
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