The boat capsized in a flooded tributary of the Kosi River in the state's Madhepura city late Thursday, the Press Trust of India news agency reported.
The dead included five girls and four boys, ages 8 to 12, a local official said. The 10th victim was not identified.
The report said rescue personnel were able to save one girl, who was admitted to a hospital in serious condition.
The victims, members of two families, were returning home after attending a function on the other side of the river.