After a midwife failed to tell his expectant mother to attend hospital immediately, the teenager was left facing a lifetime of disability.
The boy, now 17, was delivered by emergency caesarean section at the Harrow -based hospital in 1999.
His mother's first child had been born the same way, so it was a "high risk pregnancy and labour", the High Court heard on Tuesday (October 3).
On the day before the birth, the expectant mum had been admitted to hospital but had then been sent home.
Mr Justice Spencer said it was agreed that "it had been reasonable for her to be discharged".
But the boy's case focused on a telephone conversation between his mother and a midwife later the same day.
The teenager's lawyers argued that, on the basis of what she was told, the midwife should have advised the mother to return to hospital immediately.
Her failure to do so, they claimed, led to a crucial delay in the boy's delivery that left him facing a lifetime of disability.
North West London Hospitals NHS Trust denied liability for the boy's injuries while the judge acknowledged the issues in the case were "finely balanced".
The trust also argued that, even had the mother returned to hospital sooner, it would have made no difference to the tragic outcome.
Following negotiations, however, the trust on Tuesday (October 3) agreed to settle the teenager's claim for £1m.
Mr Justice Spencer said: "The court has no hesitation whatever in approving the settlement."
He added there was "clearly a significant risk of the litigation failing altogether" had the case gone to trial.
"£1m is going to make some significant difference to him and is much better than getting nothing at all," he added.
Earlier in the hearing, the boy's lawyer, Lizanne Gumbel QC, described it as an "unusual case with a number of factual issues".
After the hearing, the grateful teenager handed out thank you presents to his solicitor and barrister.
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