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Welch was born to Virgil and Deone Welch. He had two brothers—Wesley and Robert (both deceased) and a sister, Irene.
Welch's first memories were of Buck Creek, Iowa, where his father had been appointed minister of a very small church. The family moved to Fayette, Iowa (population then was about 900) where his father was Professor of Bible Studies at Upper Iowa University, at that time a very small school of about 300 students. Virgil taught Bible and spent some time traveling with Claude around the state recruiting students. Claude was unique in winning the top statewide student awards in five subjects while in high school.
Welch graduated from Upper Iowa University with a B.A. in history, received his B.D. and M.Div. from Yale Divinity School in 1945, and his Ph.D. from Yale in 1950. His doctoral dissertation was later published as In This Name: The Doctrine of the Trinity in Contemporary Theology (Scribners, 1952) and is considered to be a seminal 20th century work on the Trinity.
Welch taught at Princeton from 1947–1951 and Yale University Divinity School from 1951-1960. He served as Director of Graduate Studies in Religion at Yale from 1954-1955. From 1956 to 1957 he engaged in research on Karl Barth at the University of Heidelberg on a Fulbright scholarship.
He was appointed Berg Professor of Religious Thought and Chairman of the Department of Religion at the University of Pennsylvania in 1960. In 1964 Welch was appointed Associate Dean of the College of Arts and Sciences. Another Fulbright was spent in Neuchâtel, Switzerland, from 1964-1965 where he did work on the first volume of his survey of 19th century Protestant thought. In 1969, Welch was appointed the Director of the Study of Graduate Education in Religion by the American Council of Learned Societies. He was elected President of the American Academy of Religion in 1970. In the fall of 1971, he was appointed Dean and Professor of Historical Theology of the Graduate Theological Union in Berkeley, California, and was appointed president of GTU the following year. He retired from teaching in 2006 when he moved to Illinois.
Welch's research interests focused primarily on modern theology and philosophy of religion, 18th to 20th centuries, and theology and natural science. He claims to have intended to become a systematic theologian and spend more time on the 20th century, but maintained that one cannot do 20th century theology without understanding the 19th century first, or the 19th without the 18th.
- Protestant Thought in the Nineteenth Century, 2 vols. (Yale, 1972, 1985)
- "Nineteenth Century: An Overview," in Oxford Companion to Christian Thought (Oxford, 2000)
- (with John Dillenberger) Protestant Christianity, Interpreted Through Its Development, 2nd edition (Macmillan, 1988)
- The Reality of the Church (Scribners, 1958)
- Graduate Education in Religion: A Critical Appraisal (Montana, 1971)
- In This Name: The Doctrine of the Trinity in Contemporary Theology (Scribners, 1952)
- The Boom in Religion Studies
-  Fred Sanders: Claude Welch, 1922–2009
-  GTU: In Memoriam: Claude Welch