Cardinal Mejía was born on 31 January 1923 in Buenos Aires (Argentina). He was ordained on 22 September 1945 following studies at the Seminary of Villa Devoto in Buenos Aires. He worked in St Rose of Lima parish in Buenos Aires before going to Rome for further studies. He obtained a doctorate in theology from the Pontifical University of S Tommaso (Rome) and a licenciate in biblical science from the Pontifical Biblical Institute (Rome). He taught Sacred Scripture, Biblical Greek and Hebrew and archeology at the Catholic University of Argentina. He was a director of the Catholic journal "Criterio" from 1956-1977. He undertook further studies at the Biblical School of Jerusalem and was guest professor at the Ecumenical Institute of Higher Theological Studies in Israel. He attended Vatican II as a peritus from 1962-1965 and was director of the Commission on Ecumenism of the Archdiocese of Buenos Aires in 1966. In 1967, he was Secretary of the Deparment of Ecumenism of CELAM and then President of the Executive Committee of the World Catholic Federation for Biblical Apostolate from 1969-1972. He was appointed Secretary of the Pontifical Commission for the relations with Judaism in the Secretariat for Christian Unity in 1977.
He was appointed Vice-President of the Pontifical Commission for Justice and Peace on 8 March 1986 and assigned the titular diocese of Apollonia. He was consecrated in Rome on 12 April 1986 by Cardinal Etchegaray, President of the same Pontifical Council. He was raised to the dignity of Archbishop and appointed Secretary of the Congregation for Bishops on 5 March 1994. He was made Archivist and Librarian of the Holy Roman Church on 7 March 1998.