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Russia, Olympic silver medallists in 2008, led 10-7 after the first half and, although France fought back to 14-14, the Russians would not be denied.
Norway, gold medallists at Beijing 2008 and London 2012, beat the Netherlands 36-26 to clinch the bronze medal.
There could still be Olympic glory for France as their men's side - Olympic champions in 2008 and 2012 - play Denmark for the gold medal on Sunday.
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Russia beat France 22-19 to win women's handball gold at Rio 2016.