After suffering a period of bad health, in November 1956 Matthew Beovich requested that he receive an
auxiliary bishop.
Pope Pius XII appointed Gleeson to the position, and he was
consecrated by Beovich on 21 May 1957, In July 1964, Gleeson was appointed coadjutor archbishop, in effect naming him as Beovich's successor. Beovich retired on 1 May 1971, making Gleeson the first South Australian priest to become Archbishop of Adelaide. On 20 March 2000, Archbishop Gleeson died as the result of a heart attack. His funeral was held at St. Francis Xavier's Cathedral. The Catholic school
Gleeson College is named in his honour. == Notes ==