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The marine mammal was only nine days old when it died at the Varna Dolphinarium, according to local reports.
The dolphin is thought to have died during the show in front of a packed crowd. After the incident the spectators said that the arena was cleared quickly by staff, who ushered the crowds out of area.
An audience member told Bulgarian TV show BTV Novinite: "There was a disturbance, the dolphins stopped playing and performing tricks."
Whilst it isn't immediately clear what the dolphin died of, a lot of people have criticised the dolphinarium for the conditions.
The water park has dolphin shows four times each day during the during summer season.
All of the dolphins are forced to perform throughout the day for the hordes of tourists who visit the popular park.
Despite only being a small matter of days old, the young dolphin was forced to perform alongside the older dolphins, it has been claimed.
People on social media have heavily attacked the park after the death of the baby dolphin, even going so far as to call them a 'torturer' and accusing them of overworking the cetaceans.
Yavor Gechev, a representative from animal protection organisation Four Paws told BBC News that the death count at the park is higher than average.
He also said that a seal and five other dolphins have died at the Varna Dolphinarium over the past five years.
He said: "This means categorically that conditions don't meet even the minimal standards for keeping such animals.
"The animals are struggling there, they are not surviving."
Officials from the park have responded in comments made to local media. In these comments they have claimed that the death didn't actually take place in front of a crowd.
A biologist at Varna, Tsvetan Stanev, said: "A baby dolphin taking part in performances nine days after it was born - this has never happened in this dolphinarium!"
What is certainly true is that dolphins that are born in captivity have a massively higher mortality rate than their counterparts born in the wild.
In fact, between 50 per cent and 75 per cent of those born in facilities such as the Varna Dolphinarium die within the first year of their lives.
It is also thought that animals with high intelligence such as whales and dolphins suffer mental anguish in captivity.
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