Pc Gordon Semple vanished shortly after leaving the the five-star Shangri La hotel in The Shard skyscraper, which is next to London Bridge.
Officers conducting a forensic search of a property in Southwark discovered a body, which has not yet been formally identified.
However, detectives arrested a 49-year-old man at the scene on suspicion of murder and informed Pc Semple's family of the development.
Scotland Yard said: 'We are unable to make formal identification at this stage, but this is devastating news for all those involved in the search for Gordon.
'His family are being supported by specially trained officers and his colleagues have been briefed.
'Police were called at 13:07hrs, today, Thursday, April, 7, to an address on the Peabody Estate, Southwark Street, SE1.
'Officers attended the address where human remains were discovered.
'Our work at that scene is crucial to ensure we capture all available evidence and this may take some time.
'At this stage we will not speculate any further on what may have happened or possible motive.
'A post-mortem will be held in due course, and until that has taken place we will not speculate on the cause of death.'
The Metropolitan Police are expected to be releasing a further statement tomorrow at 8am.
He was last seen on Friday, April 1, leaving The Shard.
A CCTV camera recorded Pc Semple on Great Guildford Street at 3pm on Friday.
He was reported missing by his male partner later that evening when he failed to return to home in Greenhithe, Dartford, Kent.
Earlier, specialist divers searched the River Thames in an effort to find any clues to locate Pc Semple.
Eyewitnesses said they saw teams of forensic officers searching the communal bins in the housing estate which dates from 1876.
Pc Semple was a crime prevention officer attached to Westminster City Council.
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