The boy, named Kiron, was born by Cesarean section on August 25, weighing 12 lbs 1 oz.
He died at two days after developing a fever and breathing difficulties on Wednesday evening.
The boy had been placed under police protection when more than 150,000 people gathered outside the clinic in the town of Keshobpur where he was being treated.
He was moved to a hospital in the neighbouring city of Jessore, but his parents ignored doctors' advice to seek further treatment in the capital.
"We wanted to refer him to a hospital in Dhaka but the family was so poor that they could not afford to take him there, so they took him home where he died," KS Alam, one of the doctors treating the boy, said.
"It was a very unusual case. The boy had one body but two complete heads."