PC Gordon Semple, 59, was last seen on Friday April 1, and had left his home in Greenhithe, Dartford that morning to go to work in Westminster.
Scotland Yard said human remains have been discovered and a 49-year-old man has been arrested on suspicion of murder.
A statement from the Metropolitan Police said: "A 49-year-old man was arrested at the address on suspicion of murder and is in police custody.
"This is obviously a significant development, and we have informed Gordon's family.
"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, 7 April, 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."
Anyone with information is asked to call the incident room on 020 8721 4961, or to remain anonymous, call Crimestoppers on 0800 555 111.