The Nick Williams-trained 12-year-old pulled up sharply before the final flight at the Edinburgh Gin Handicap Hurdle.
Before jockey James Reveley had a chance to dismount, the horse fell to the floor.
Course clerk Anthea Morshead said: "It was very sad, he suffered an internal bleed and collapsed and died."
Reve De Sivola won the Long Walk Hurdle at Ascot for three consecutive years between 2012 and 2014.
He also won the Challow Hurdle at Newbury as a novice, as well as the Champion Novice Hurdle at Punchestown, with career win and place prize-money of more than £600,000.
The three-time Long Walk Hurdle winner Reve De Sivola has died after collapsing at Kelso on Saturday.