The first woman's mother, a Catholic former midwife, had earlier this year won a legal battle after doctors said the pregnancy should be terminated.
Lawyers involved in the case said on Tuesday that the woman, who is in her 20s but has the mental age of a child aged between six and nine, had given birth and the baby was doing well.
The case hit the headlines in June after a judge ruled that the woman should have an abortion.
Mrs Justice Lieven had analysed evidence at a hearing in the Court of Protection, where issues relating to people who lack the mental capacity to make decisions are considered, in London.
She concluded that termination was in the woman's best interests.
That decision was overturned by Court of Appeal judges who decided that the woman should be allowed to give birth.
Bosses at an NHS hospital trust responsible for the woman's care had asked Mrs Justice Lieven to let doctors perform an abortion.
Three specialists, an obstetrician and two psychiatrists said a termination was the best option because of the risk to the woman's daughter's psychiatric health if pregnancy continued.
They said the woman would not be able to care for a child.
But the woman's mother had been against termination and said she could care for the child.
A council social worker who worked with the woman had also said the pregnancy should continue.
Lawyers who represented the woman also said she should be allowed to give birth.
Judges have heard that the "circumstances of the conception" are "unclear" and that a police investigation is ongoing.
They have ruled that the woman, who lives in the London area, cannot be identified in reports of the case.
They also say the NHS hospital trust which asked for a decision, and the council which employs the social worker, cannot be named because publication of their names might create an information jigsaw which could lead to the woman's identity being revealed.
At the same time, a separate but similar case has led to the opposite conclusion at the Court of Protection, surrounding a woman in her twenties who has the mental age of a toddler.
This woman is 12 weeks pregnant and specialists are preparing to terminate her pregnancy as she lacks the capability to make decisions. She has been fostered and her foster parents are Christians.
Mr Justice Williams said medics could restrain the woman in order to administer a general anaesthetic if necessary this week.
The judge said the woman, who lives in the north of England, could not be identified in media reports.
He also said the names of the hospital and council involved could not be revealed in case that information created a jigsaw which pointed to the woman's identity.
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