Father Martin Xavier Vazhachira had last been seen in St John The Baptist RC Church in Corstorphine at around 1pm on Tuesday.
The alarm was raised by parishioners after the 33-year-old did not turn up for mass on Wednesday morning.
A body was discovered in the West Barns area of East Lothian on Friday at 4pm.
It has since been identified as that of the priest.
Archbishop Leo Cushley 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.
“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.”
Fr Vazhachira was a native of Kerala in southern India and was ordained a priest of the Carmelites of Mary Immaculate Order in 2013.
He arrived in Scotland in July 2016 to undertake postgraduate studies at Edinburgh University and served at parishes in Falkirk before being appointed to Corstorphine in October 2016.
Archbishop Cushley will be at the church on Saturday and Sunday to offer mass and support parishioners.
A Police Scotland spokeswoman said: “The family of Father Martin Xavier Vazhachira has been informed of this development.”