Strathclyde Fire and Rescue were called to a ground-floor flat yesterday morning.
Neighbours said the occupier was inside the flat in the St Leonards area of the South Lanarkshire town.
Eight other residents from the four-storey block were led to safety and three were treated in hospital for smoke inhalation.
Station commander Jim Anderson, of Strathclyde Fire & Rescue, said the crews were "working in thick smoke and extreme heat".
Police said a 77-year-old woman died in the fire. Investigations are ongoing but officers said circumstances surrounding the blaze were not thought to be suspicious.