Conway was ordained to the priesthood on 23 June 1957 and has spent the greater part of his priestly career as a teacher in St. Malachy's College, Belfast.[1] In 1983 he succeeded Patrick Walsh as president of St Malachy's College. One of his most significant, if tangential, achievements was to preserve the O'Laverty collections of historical papers and books that today forms an important resource for the wider history of Belfast.[2]
He served as president until 1994 and then after recuperation from illness was, in 1995, appointed parish priest of Strangford in County Down.[3]
