Born the son of a semi-literate but wealthy cattle farmer in the Ixopo district of Natal in 1919, Conco rose to become Luthuli's second-in-command in the Natal African National Congress. He attended Marianhill College, where he distinguished himself as a student. After a year of premedical studies at Fort Hare, he studied medicine at the University of the Witwatersrand, graduating in 1948.
In the company of Arthur Letele, Lionel Majombozi, and William Nkomo, who were then also medical students, Conco attended some of the foundation meetings of the Youth League. He did not develop a serious interest in politics, however, until several years of medical practice had impressed him with the depths of rural poverty in Natal.
He joined the ANC in 1950 and was active in the 1952 Defiance Campaign. In 1954, he was elected national treasurer of the Youth League and for a time also served as acting treasurer of the ANC. Several months after the 1955 Congress of the People, at which Conco played a prominent role, the government issued a banning order restricting him to the Ixopo district. As a first-string accused in the Treason Trial, however, he spent most of the next five years in the Transvaal, commuting to see his patients on weekends whenever he could.
His friends praised him as a man of quiet, self-sacrificing devotion to the African cause and a committed adherent of non-violence. On his release from the Treason Trial in 1961 he moved his medical practice to Swaziland. In 1968 he moved to London, where he became involved in the work of the Lutuli Memorial Foundation. In the early 1970s he was a psychiatry resident at the University of Toronto, Canada.