"Fire broke out in one of the tents when some priests were cooking. The tent caught fire and spread swiftly," said R S Rathore, an administrator at the festival in the northern state of Uttar Pradesh.
"Twenty tents were gutted and one priest has died," he told AFP.
Rathore said that strict instructions had been issued beforehand against lighting fires in tents at the festival, which at its height last Sunday drew a crowd of 30 million people, making it the world's largest gathering.
Thirty-six people were killed on Sunday night in a crush at a train station in the town of Allahabad as pilgrims headed home from the Kumbh Mela.
The 55-day festival on the banks of the River Ganges has seen tens of millions of Hindus take a dip in the sacred waters in a bid to cleanse them of their sins.
The Kumbh Mela, which ends in March, takes place every 12 years in Allahabad while smaller events are held every three years in other locations around India.
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