The woman gave birth by Caesarean section at the Recoletas Hospital.
C section is the usual method in such rare cases.
The unnamed woman had undergone fertility treatment in the United States, Spanish media report. The hospital has posted a video clip of the Caesarean delivery.
In 2012, the woman gave birth to a girl, who was later taken into care by social services, amid welfare concerns.
According to the Spanish daily El Pais, social workers said the girl was being brought up isolated, poorly clothed and with bad personal hygiene.
No decision has been taken yet about the mother’s twins. They and the mother are reported to be in good health in hospital. The boy weighs 5.3lbs and the girl 4.8lbs.
The delivery went without complications.
In recent years, El Pais reports, two other Spanish women in their 60s have given birth to healthy babies.
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