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Property manger Craig Paterson said thousands of dollars of damage had been done.
As well as damaging historic paintings and other irreplaceable property, the offenders broke into a flat on site.
He said an elderly man lived there and kept watch on the building for the diocese.
"Fortunately, he wasn’t home when the building was broken into.
"They trashed the flat and left the bath on and flooded the place. The damage is significant.”
He said the building was locked and alarm on at the time. It was also insured.
Senior Sergeant Craig Dinnissen, of Dunedin, said, on the same day, a student flat in Cumberland St, North Dunedin, was burgled.
"We are aware that when times get tough, people start going window-shopping at residential addresses and commercial premises."
Snr Sgt Dinnissen said there had not yet been a noticeable increase in burglaries.
"I think it’s probably because a lot more people are at home.
"We hope that there won’t be an increase.”