Four hours after it began, the fire brigade said the blaze was under control.
The brigade said 30 residents had been rescued from the building "in extremely difficult circumstances."
Police said two people had died -- a 3-week old baby and a 7-year-old child.
The London Ambulance Service said 16 people had been taken to a local hospital, most suffering from smoke inhalation. At least three were in serious condition.
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