The jail was "largely destroyed after IS planted explosives inside and around it", 10 days after the jihadists seized Palmyra from regime forces, the Syrian Observatory for Human Rights said.
IS supporters posted pictures on Twitter purporting to show the infamous prison being blown up.
The prison was the site of a massacre in 1980 in which hundreds of inmates were killed.
It became notorious throughout Syria as a symbol of the brutality of the regime of former president Hafez al-Assad and his son and successor, Bashar.
Regime opponents praised the jail's destruction on social media.
"IS has wiped out evidence of the crimes of the Assad clan by blowing up the infamous Palmyra prison," said Syrian opposition member Mohammad Sarmini on Twitter.
"Palmyra prison bears witness to the crimes of the century," an anti-Assad activist tweeted.
The Observatory said government forces relocated inmates held at the jail before IS overran the historic city of Palmyra, which houses priceless UNESCO-listed ruins and ancient artefacts.
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