Clarence Elwell was born in the Newburgh area of Cleveland on February 4, 1904. As a boy, Clarence served Mass for his assistant pastor, Father Michael J. Ready at Holy Name Parish, where he also attended elementary and high school. He began attending St. Ignatius College and studying medicine, but he switched to St. Mary Seminary in Lakeside, Ohio, after his second year of college. He also studied at the Canisium, the seminary at the State University of Innsbruck, Austria. Clarence was ordained in Innsbruck by Bishop Sigismund Waitz, Auxiliary Bishop of Brixen.
Fr. Elwell returned to Cleveland as an assistant pastor and teacher. He also served as assistant superintendent of the Diocese of Cleveland's schools. In 1934, Fr. Elwell earned a master's degree in education from Western Reserve University, and in 1938 he received a doctorate in education from Harvard University. Upon returning to Cleveland, Fr. Elwell was appointed director of Catholic high schools. In 1946 he was named superintendent of schools for the Diocese of Cleveland.
Fr. Elwell was named a monsignor in 1949. On November 5, 1962, Pope John XXIII named Msgr. Elwell Titular Bishop of Cone and Auxiliary Bishop of Cleveland. He was consecrated in St. John Cathedral in Cleveland on December 21, 1962.
Bishop Elwell was appointed Bishop of Columbus on May 29, 1968 and was installed in St. Joseph Cathedral on August 22 of that year. He continued the work of implementing the reforms of Vatican II that were started by Cardinal Carberry, including giving permission for Saturday evening Mass and other liturgical changes.
Bishop Elwell took a strong interest in Catholic schooling and worked with his fellow bishops in Ohio to foster opportunities for families to send their children to Catholic schools. During his episcopate, three high schools were created and built: Tuscarawas Central Catholic in Dover, Fisher Catholic in Lancaster, and Bishop Rosecrans in Zanesville. He closed St. Charles Seminary and converted it to the college preparatory school that it is today.
He also founded Resurrection Cemetery in Columbus, and established St. Peter Parish in the Worthington area, as well as the Diocesan Sisters' Council and the Diocesan Pastoral Council. He moved lay people into positions of trust, such as director of cemeteries, editor of the diocesan newspaper, and superintendent of buildings. Additionally, he firmed up diocesan finances by expanding the Diocesan Development Office, the Parish Aid Fund, and the diocesan self-insurance program.
Bishop Elwell died unexpectedly February 16, 1973, in his apartment in the diocesan Chancery. Bishop Elwell is buried in St. Joseph Cemetery in Columbus.