ROME—The Vatican said Thursday it had convicted an Italian businessman of laundering money through the Vatican bank, in what it said was its first conviction under a 2010 law and a sign of progress in financial reforms.

The Vatican court on Dec. 17 sentenced Angelo Proietti, a builder who had done construction work for various Vatican offices, to 2½ years in prison, and to the loss of more than €1 million ($1.14 million) in an account he held at the Vatican bank.

Mr....