The Lyon court said Alexei Kuznetsov should be sent back to Russia on condition that he was not subjected to forced labour, and that he could receive visits from the French authorities.
Kuznetsov was arrested close to St Tropez in the summer of 2013 in possession of passports in five different names.
He stands accused by the Russian authorities of embezzling some 14 billion rubles, more than €400,000 million at the time, when he was finance minister for the Moscow region between 2005 and 2008.
The money was allegedly laundered in property abroad, including France.
Kuznetsov apparently fled Russia when a business associate was murdered. He then resurfaced in France.
Russia later issued an international arrest warrant for him.