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An overcrowded railway station footbridge buckled and a railing collapsed, sending some people slipping down the stairs and triggering the stampede, a top state government official told Reuters, not wishing to be quoted by name.
"I can confirm that 18 people have died and 13 have been injured," said the official.
Once every 12 years, tens of millions of pilgrims stream across India to the small northern city of Allahabad for the Maha Kumbh Mela, or Grand Pitcher Festival, at the point where the Ganges and Yamuna rivers meet a third, mythical river.
In the two months from the start of the festival in January, officials believe as many as 100 million people will have passed through a temporary city that covers an area larger than Athens on a wide sandy river bank.
Hindu holy men and pilgrims bathe in the sacred Ganges to wash away lifetimes of sins. Sunday was believed to be the most auspicious day of the festival.
Prime Minister Manmohan Singh expressed shock at the tragedy and promised financial assistance to the families of the dead.