Conus was born in Moscow, the son of the pianist Eduard Conus. His brothers were the musicians Georgi Conus and Lev Conus. All three brothers studied in the Moscow Conservatory; among their teachers were Sergei Taneyev and Anton Arensky. Following the Russian Revolution he emigrated to France, where his son, Serge Conus, was born. Another of his sons, Boris, married the daughter of Sergei Rachmaninoff. Julius Conus returned to Moscow in 1939, dying there in 1942. His compositions include a concerto for violin which has been recorded by, among others, Jascha Heifetz.