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The Dnipropetrovsk branch of the Emergency Situations Ministry said the casualties were on the bus, 38 people died on the spot and five more in the hospital. The victims included a child and two teenagers.
Some of the injured are critical conditions and the death toll could continue to rise.
President Viktor Yanukovych conveyed his condolences to the relatives of the victims and said a committee headed by Deputy Prime Minister Andriy Kluyev had been set up to investigate the cause and help the families.
A statement on the presidential website said Kluyev had already left for the scene of the tragedy.
"The head of state has ordered law enforcement agencies to carry out a detailed investigation to establish those responsible for the accident," the statement said.
Yanukovych later declared a national day of mourning Wednesday.
Prime Minister Mykola Azarov ordered his government to pay the family of each of the dead passengers 100,000 hryvna ($12,600).
He also instructed transport officials to install automated crossing gates at all railway crossings to prevent cars, buses and trucks from ignoring signals.
Ukraine has a bad safety record due to the poor condition of many roads and drivers who regularly flout the rules. Officials said this was the deadliest auto accident in the country's history.