Together with an accomplice, the man had demanded that bank employees hand over money, threatening the staff with box cutters.
Shortly after issuing the threats however, he collapsed on the scene.
The thief's companion reportedly attempted to revive him - and then to drag him out of the bank - but gave up, fleeing with around 5,000 euros stolen from the tills.
Paramedics who arrived with police shortly after also tried to resuscitate the collapsed man, but were unsuccessful.
The incident took place shortly before 11am at a branch of Banca Popolare in Lajatico, a small hamlet in Tuscany.
On Tuesday morning, the thief's accomplice was still on the run. According to local paper Pisa Today, the pair had strong Campanian accents, and the deceased man was aged around 40. Police are attempting to identify him from his fingerprints.
Lajatico, home to around 1,000 people, is situated 40km southeast of Pisa and roughly 50km southwest of the region's capital, Florence.