Her son Gabriel and daughter Maria de la Cruz have been taken in care by social services after an investigation found there were “signs they could be at risk”.
Íbanez, a retired civil servant from a village near Burgos in northern Spain, also has a six-year-old daughter, Blanca, who was taken into care three years ago and now lives with a relative in Canada.
Last month she spoke of her fear that her newborns would be taken away from her.
“They're so defenceless, so delicate. I'm a bit afraid for them. I just ask God that they don't fall into the hands of social services,” she said in a revealing interview with El Pais.
Madre de mellizos a los 64 años; así se siente Mauricia Báñez un mes después de dar a luz https://t.co/FoRgpbJurM— EL PAÍS España (@elpais_espana) March 14, 2017
She had explained that even her own family had been against her seeking fertility treatment, which she underwent in the US using donor eggs because clinics in Spain refused to assist her on account of her age.
After taking early retirement from the foreign office after being diagnosed with a paranoid personality disorder a decade ago, her own sister sought a court order to prevent her going abroad for treatment.
Ms Ibanez's has not commented publicly since her children were taken from her but her lawyer Juan Carlos Saiz told El Pais: "She is devastated and very traumatized.
"The first time this happened was very painful for her. This time it's even worse."
News of the birth, which was announced by the hospital without revealing the identity of the mother, sparked furious debate in Spain over the ethics of becoming a mother well past the normal age of retirement.
Ibañez is one of the oldest women in the world to give birth, but is still two years younger than the current holder of the record, fellow Spaniard Maria del Carmen Bousada Lara, who also gave birth to twins.
Bousada, who was born on January 5th 1940, gave birth by Caesarean section to twin boys, Christian and Pau, a week before her 67th birthday at the Sant Pau hospital, Barcelona, Spain on December 29th 2006.
Unfortunately, her sons were orphaned at two years of age when their mother died of cancer.
Another Spanish woman, Lina Álvarez, from Lugo in Spain's northwestern region of Galicia, made headlines last October, when she gave birth to a healthy baby daughter at the age of 62.