The girl’s lawyers claimed her delivery was delayed after medics at Oxford University Hospitals NHS Foundation Trust failed to see the warning signs.
The trust had argued the damage to her brain was done before her mother went into labour.
It earlier agreed to settle a clinical negligence claim.
The girl, who was born at Oxford’s John Radcliffe Hospital, is now nine years old.
She will receive a lump sum of £1.3m, and £50,000 a year, index linked, for the rest of her life to pay for her care.
Her legal team claimed doctors failed to see she was suffering distress in the womb.
‘All or nothing’
The court heard her brain had been starved of oxygen, resulting in cerebral palsy. She also has severe learning difficulties, gravely impaired vision, and epilepsy.
Her barrister, Elizabeth-Anne Gumbel QC, said it was an “all or nothing” case.
She said the settlement reflected the risks of litigation and represented “about 30%” of the full value of the damages claim.
But the trust had denied the birth was “mismanaged”, and NHS counsel Margaret Bowron QC said it was “very pleased” with the compromise.
She said the girl “had more thrown at her in her short life than anyone has a right to expect”.
She had coped bravely with the support of her parents “who are obviously deeply, deeply, fond of her and she of them,” she added.
Deputy Judge Heather Williams QC said: “I am content to approve the settlement, which seems to me very sensible, and the parties are to be congratulated on achieving a resolution.”
She called it a “very tragic” and “difficult” case.