Curt Hugo Johannes Weibull was born in Lund, Sweden. He was a member of the noted Swedish Weibull family. He was the son of professor Martin Weibull (1835-1902) and brother of historian Lauritz Weibull (1873-1960). He and his brother both attended the University of Lund. 
Curt Weibull was a professor of history at the University of Gothenburg from 1927-1953 and its rector from 1936 to 1946. In 1928 he and his brother, Lauritz Weibull, initiated the periodical Scandia. Together they are known for having introduced a critical theory of history to Swedish historical research, inspired by German historian Leopold von Ranke (1795-1886). Weibull was an important mentor to noted Swedish historian Erik Lönnroth (1910-2002) who further developed the methods of evaluating sources.
His most important and acclaimed work is a criticism regarding the interpretation and the ahistoricism of the Gesta Danorum by the 12th century Danish historian Saxo Grammaticus. This piece was named: Saxo. Kritiska undersökningar i Danmarks historia från Sven Estridsens död till Canute VI (Saxo. Critical studies in Denmark's history from Sven Estriden's death to Canute VI), and was rather controversial at the time, as it revealed the vague basis of Denmark's older history of the time.
In 1991, when he was 105, his last work was published: an article in a book celebrating the 100th anniversary of the University of Gothenburg. That probably makes him the oldest historian in the world to have a new study published while still alive. An anecdote tells that when a Danish historian was counter-criticizing parts of Weibull's Ph.D. thesis (from 1916) on Saxo Grammaticus in his own thesis (believing Weibull to be dead since this was after his 100th birthday) Weibull appeared in the public disputation angrily defending his work. During the late 1970s, while lecturing about his life and his research to younger students, he had humorously remarked about a Danish professor who had criticized his own thesis when it appeared: "I haven't replied in depth to the criticism of the professor. But it's not too late, now is it?"