Photos of the public event were posted on Twitter by several jihadi accounts, including "Jehad News", which said the thief in the northern town of Maskanah had admitted his crimes "and also asked that his hand be cut off to cleanse his sins".
It was not immediately possible to verify the accounts. One photo showed a blindfolded man with his hand being held on a table while several militants watched, one brandishing a large knife.
A subsequent photo showed the man with his hand severed.
The punishment was inflicted by the Islamic State in Iraq and the Levant (ISIL), an al Qaeda splinter group that is widely considered the most radical of the groups fighting President Bashar al Assad.
The group, which is mainly composed of foreign fighters, is also engaged in a violent struggle with rival Islamist rebels.
ISIL controls much of rebel-held Syria in the north. In the city of Raqqa, it has demanded that Christians pay a levy in gold and curb displays of their faith in return for protection.
It has also banned Christians from owning weapons and from selling pork or wine to Muslims or drinking wine in public.