A tyre on the bus burst as it was crossing the bridge over the Kuenga river in eastern Siberia’s Zabaikalsky region.
The vehicle, which was travelling from Sretensk to Chita, skidded off the road and onto the ice.
Two helicopters with medics were dispatched to the scene.
Russia’s Investigative Committee, which probes serious incidents, said it had opened a criminal inquiry into a possible violation of traffic safety rules.
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However, the number of road deaths has gone down in recent years, to around 20,000 per year.