The world-famous revolving sign will be retained at the new building, as well as the Eternal Flame and Roll of Honour, which are currently located at New Scotland Yard in Victoria.
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The Marsh & Parsons poster - captioned "a charming period property with a modern extension" - soon prompted criticism.
The track will be demolished to make way for a housing development and a new Stadium for AFC Wimbledon.
Just four authorities (Hammersmith & Fulham, Hillingdon, Newham and the City of London) have chosen to freeze council tax altogether.