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Sensi was born in Cosenza, Italy as the sixth of ten children of a prominent local politician. He was ordained a priest in December 1929 at the age of 22 and by 1934 he was part of the diplomatic service fo the Holy See - where he was to remain for the rest of his life as a priest. He worked in minor roles in many nunciatures before he became an observer for the Holy See with the United Nations in 1953.
On 21 May 1955 Pope Pius XII appointed him Titular Archbishop of Sardes and Apostolic Nuncio to Costa Rica. He remained in Costa Rica until 1957 when he was appointed Apostolic Delegate to Jerusalem and Palestine. Pope Paul VI appointed Archbishop Sensi on 10 May 1962 to be the sixth ever Apostolic Nuncio to Ireland. He remained in Ireland until 1967 when he was transferred to Portugal.
In 1976 was appointed to the Roman Curia and on 24 May 1976 he was created and proclaimed Cardinal-Deacon of Santi Biagio e Carlo ai Catinari by Pope Paul at the age of 69. He took part in both of the conclaves of 1978 that elected Pope John Paul I and his successor Pope John Paul II.
After ten years as a Cardinal Deacon he took the option of becoming a Cardinal-Priest on 22 June 1987, becoming Cardinal-Priest of Regina Apostolorum.
He died at the age of 94 in 2001.