Pompedda was born in Ozieri in Sardinia. He studied at seminaries in Sassari and Cuglieri and was ordained a priest in Rome on 23 December 1951. He obtained a doctorate in theology from the Pontifical Gregorian University, and a doctorate in utroque iure from the Pontifical Lateran University. He became an officer of the Roman Rota, the general appellate court of the Catholic Church, in 1955, serving as a defender of the bond. He joined the tribunal as an auditor in 1969 and served in that capacity until he became Dean of the Roman Rota in 1993. He was appointed Prefect of the Supreme Tribunal of the Apostolic Signatura, the highest tribunal in the church, in 1999. He resigned this position, as is customary, upon reaching the age of 75, on 18 April 2004, and his resignation was accepted on 27 May 2004. Pompedda was appointed titular Archbishop of Bisarcio on 29 November 1997 and consecrated on 6 February 1998. He was proclaimed a Cardinal-Deacon of Santissima Annunciazione della Beata Vergine Maria a Via Ardeatina by Pope John Paul II on 21 February 2001. He was one of the cardinal electors who participated in the 2005 papal conclave that selected Pope Benedict XVI. It was suspected that he may have been the source of information about the conclave published in the Italian journalist Lucio Brunelli in the journal Limes in September 2005; many also suspected that the information was false. Cardinal Pompedda also served as president of the appellate court of Vatican City and, from 1999, as president of the Court of Cassation of Vatican City. He is credited with drafting Universi Dominici Gregis. Pompedda taught canon law at the Pontifical Gregorian University. He was also involved in the revision of the code of canon law in 1983, the first substantial revision since 1917. He was also the principal editor of Universi Dominici Gregis, the apostolic constitution of Pope John Paul II published in 1995, which set the rules for the sede vacante and papal conclave. He supported proposals to make it easier to obtain an annulment of marriage, and gave several public lectures in 2003 outlining the changes. He retired before the changes were implemented, and the final document, Dignitas connubii, did not go as far he had expected, reaffirming existing doctrine instead. Pompedda was appointed as the Cardinal Grand Prior of the Sacred Military Constantinian Order of Saint George by order of the Grand Master, the Duke of Castro. In November 2003, he became the Ecclesiastical Counsellor of the Order by appointment of Pope John Paul II. This position continued under Pope Benedict XVI. Cardinal Pompedda died in Rome of a brain hemorrhage. He was buried in a tomb in the cathedral in Ozieri. On 18 February 2010, his remains were reinterred in a specially constructed sarcophagus in the cathedral of Ozieri at the chapel of the Most Holy Sacrament. The requiem mass was celebrated at 5 pm, presided by Bishop Sergio Pintor of Ozieri. At the end of the mass, Mgr Giuseppe Sciacca, judge of the Roman Rota and a long time collaborator of Cardinal Pompedda, read a special message sent for the occasion by Tarcisio Cardinal Bertone, S.D.B., secretary of State, in the name of Pope Benedict XVI.
