"I beg you to release him immediately, respecting his integrity and ending this painful wait as soon as possible."
Local media said Mr Diaz would undergo a medical examination before being returned to his family.
Authorities said that he appeared to be in a good state of health, with no signs of mistreatment.
According to El Tiempo newspaper, there were tearful scenes of celebration in the neighbourhood where the couple lived after the news broke of his release.
The ELN is Colombia's main remaining active guerrilla group and has been fighting the state since 1964.
The group said a week ago it would free Mr Diaz but his liberation was delayed amid a back-and-forth between the group and the government.