Victoria Burnett, The New York Times

MEXICO CITY — Cardinal Jaime Ortega y Alamino, the longtime leader of the Roman Catholic Church in Cuba and a savvy political operator who aided the Vatican’s secret efforts to broker détente between Washington and Havana, is stepping down, closing an era in which the church became the only institution outside the government with any sway on the island.

His successor as archbishop of Havana will be Juan de la Caridad García Rodríguez, the archbishop of the central city of Camaguey, according to a statement from the Cuban Conference of Catholic Bishops released Tuesday.

Cardinal Ortega, 79, will leave behind a Cuban church whose reach is greater than at any point since Fidel Castro swept to power in 1959. Far from the days when Catholics were marginalized and the cardinal — as a young priest — spent time in a labor camp, the church is building new places of worship, tending to the poor, offering courses for aspiring entrepreneurs and prodding the government to speed up economic reforms...