Ordained on October 28, 1916, Concha then served as a chaplain to the cathedral school, and a professor at the seminary and Our Lady of the Rosary University in Bogotá until 1918, while also doing pastoral work.

President of the committee for the 1925 Holy Year, he was also Sub-Promoter of the Faith in the informative process for the beatification of Ezequiel Moreno y Díaz, OAR, who was later canonized by Pope John Paul II on October 11, 1992.

He received his episcopal consecration on the following November 30 from Archbishop Ismael Perdomo, with Bishops Francisco Cristóbal Toro and Crisanto Luque Sánchez serving as co-consecrators, in the Cathedral of Bogotá.

The Colombian primate attended the Second Vatican Council from 1962 to 1965, and was one of the cardinal electors who participated in the 1963 papal conclave that selected Pope Paul VI.

Concha was staunchly conservative,[1][2] and so Colombia was slow in implementing the reforms of the Council[2] He even prohibited any funeral Masses from being celebrated for liberation theologian, Camilo Torres Restrepo.