Recoletas Hospital on Thursday said that the babies were delivered by caesarean section on Tuesday.
It said the twins - a boy weighing 2.4kg and a girl weighing 2.2kg, were "perfectly healthy".
Diario de Burgos newspaper reported the mother had her first child at age 58, in 2012.
It reported her daughter has been in the custody of local authorities since 2014 because she did not attend school and was not properly cared for.
The Spanish woman became pregnant a second time after seeking in-vitro fertilisation in the US.

Screen grab from a Recoletas Hospital video (Facebook) Source: Facebook
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