"We can confirm that 19 people have died. Rescue operations are on to find those still missing," Manu Maharaj, senior police superintendent, said.
Rescued people are being treated at the Patna Medical College and Hospital, said district officials.
According to eyewitnesses, the overcrowded boat was returning from a kite flying festival,
organised as part of Makar Sankranti by the state Tourism Department, when the incident took place around 6 p.m.
Rescue operations were hampered since it was already dark by the time. Six of the eight people rescued were pulled out from the river by the National Disaster Response Force, a team of which was deployed for the rescue operations.
Patna District Magistrate Sanjay Agarwal and district police officials are camping at the site.
Bihar Chief Minister Nitish Kumar has ordered a probe into the incident.
The state government also announced an ex-gratia of Rs 400,000 to the next of the kin of the victims.