Justice Cobb said it would be lawful for specialists to operate on the teenager, who is in council care, if necessary.
The judge made a ruling at a public hearing in the Court of Protection in London on Friday but embargoed reporting until the baby had been born.
Lawyers representing hospital bosses responsible for her treatment and care said the girl has given birth.
Cobb was told that the girl had a “history of sexual exploitation,” post-traumatic stress disorder and an anxiety disorder.
He said she could not be identified in media reports of the case.
Bosses at the Wrightington, Wigan & Leigh Teaching Hospitals NHS Foundation Trust, which is based in Wigan, Greater Manchester, had asked the judge to decide what moves were in the teenager’s best interests.
Specialists said they thought the teenager’s mental health difficulties made delivery by caesarean section the only “realistic” option.
They said efforts to induce had been unsuccessful because of the youngster’s problems.
Barrister Peter Mant, who led the trust’s legal team, told the judge that the youngster’s views on giving birth had changed repeatedly recently.
Mant said specialists thought it was in the girl’s best interests to have a caesarean section.
He said there was “a significant risk of stillbirth.”
The teenager, who was represented by barrister Mungo Wenban-Smith, said she wanted to deliver her baby naturally.
She told the judge that she had a strong wish to give birth with the least intervention and to be awake and aware when her baby was born.
Judges in the Court of Protection consider issues relating to people who might lack the mental capacity to make decisions for themselves.
Evidence indicated that the teenager’s capacity to make decisions “fluctuated,” the judge was told.
Cobb said doctors should make one last attempt to induce but ruled that performing a caesarean section would be lawful if necessary.
The judge embargoed reporting after lawyers raised concern about the girl recognizing herself in reports and behaving in a way which might put her and the baby at risk.
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