Fr Martin Xavier Vazhachira was last seen alive at his parish house on Tuesday afternoon and his disappearance prompted an appeal for information by Police Scotland.
Archbishop Leo Cushley of St Andrews & Edinburgh said: "The news of Father Martin Xavier's death comes as a great shock and a great sadness to all those who knew him and loved him."
Parishioners raised the alarm when Fr Martin Xavier didn't arrive for Holy Mass on Wednesday morning. A post-mortem will be carried out to ascertain the cause of death. His family has been informed.
Fr Martin Xavier, who was born in the southern Indian state of Kerala, had been serving as the administrator of St John's parish in Corstorphine.
Archbishop Leo Cushley added: "Our thoughts and, more importantly, our prayers are with him and with all his loved ones in both Scotland and India. May he rest in peace."
Ordained a priest of the Carmelites of Mary Immaculate in Kerala in 2013, Fr Martin Xavier arrived in Scotland nearly a year ago to study at Edinburgh University.
Before his appointment to Corstorphine in October last year, Fr Martin Xavier also served in the parish of Falkirk.
Archbishop Leo Cushley is visiting the parish of St John's in Corstorphine on Saturday and on Sunday to offer Holy Mass and to support and comfort parishioners as they face the news of Father Martin Xavier's death.