ISIS fighters packed explosives in the Baal Shamin temple and blew up the shrine on Sunday, Syria's antiquities chief said.
Ban also voiced outrage over the murder of Khaled al-Assaad, the retired chief archaeologist at Palmyra whose body was mutilated after he was beheaded on August 18.
"These barbaric acts of terror join a long list of crimes committed over the past four years in Syria against its civilian population and heritage," Ban said in a statement yesterday.
Palmyra is listed as a world heritage site by the UN cultural body UNESCO, and Ban recalled that destroying world cultural sites was a war crime. The UN chief called on world governments to "unite and act swiftly to put a stop to this terrorist activity."