The site used to be a walkway to the Royal National Orthopaedic Hospital and is between two commercial office buildings, one with a pub at the bottom.
The entrance hall will lead to an open-plan kitchen and dining room and bike store on the ground floor.
The catch? The four-storey house will be only three metres wide.
According to Guinness World Records the skinniest house in the world is the Keret house, in Warsaw, Poland.