A judge ruled Helen Coyle had proved "a causal connection" between a breach of duty by midwives and the injury sustained by Ryan, now 20, who suffers from quadriplegic cerebral palsy.
The 45-year-old mother, from Coatbridge, claimed that the brain injury her son suffered during labour at Bellshill Maternity Hospital was caused by the fault and negligence of midwives and an obstetrician.
She sued Lanarkshire Health Board in an action originally seeking £15 million at the Court of Session in Edinburgh. An agreement over the amount to be paid was reached in the event of liability being established, although details of the terms of the periodic payment order were not disclosed in Lord Tyre's judgement.
The health authority contested the action and contended there was no fault on the part of the obstetrician and that any failure by midwives did not cause the baby's injury.
Lord Tyre said: "I hold that the pursuer has proved a causal connection between breach of duty on the part of midwives, for whose negligence the defenders are vicariously responsible, and the injury sustained by Ryan. Her case therefore succeeds on the merits."