The infants, a boy and a girl, were apparently conceived via IVF treatment in the USA, and are according to the paper said to be in a 'perfect state of health, ' weighing 4lbs 8 oz and 5lbs 3 oz respectively.
But the birth has sparked controversy after it was revealed her first child, a girl, was removed from her care after being born six years ago.
The mother became pregnant following a successful IVF treatment.
Spanish newspaper Diario de Burgos reports the woman's daughter has been in the custody of local authorities since 2014 because she didn't attend school and was not properly cared for.
Enrique Martin, medical director at Recoletas hospital, said on Thursday that the babies, delivered by caesarean section on Tuesday, were "perfectly healthy". The twins are a girl who weighed 4.9 pounds and a boy weighing 5.3 pounds.
She had already conceived another daughter in 2011, but Social Services took her away because the girl was apparently lacking personal hygiene and schooling on behalf of her mother. No decision has been made yet about her twins.
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"The mother and her twins, a boy and a girl, will remain in the hospital for four or five days which is normal after a Caesarean".
According to Spanish Laws there is no limit on the maximum age for assisted reproduction.
A 64-year-old Spanish woman has successfully given birth to a pair of twins, a boy and a girl. She specializes in IVF treatments and became a symbol for many older women wishing to conceive.
BBC reported a similar case in April 2016 with an Indian woman in her 70s that underwent fertility treatment.
The world's oldest documented mother is Daljinder Kaur from India, who gave birth in 2015 at the age of 70 after two years of IVF treatment.