The building is attached to St Patrick’s Church in the town.
NIFRS said the incident was scaling back but they planned to remain at the location for a number of hours.
SDLP councillor Conor Galbraith who was at the scene said: “It doesn’t look like it could spread, with most of the fire located on the building’s roof, which appears to be caved in”
A priest had been ordained there on Sunday afternoon, “just hours before the blaze,” he said.
He added that it was sad to see the fire in “such a historical part of our town”.
Galbraith’s party colleague, Colin McGrath, described the blaze as “difficult to watch”.