It will be the second hotel in the capital for PIC, which earlier this year opened the 85-bedroom Bulgari hotel in Knightsbridge, which is managed by Marriott.
PIC was founded by Spanish businessman Rafael Serrano in 2009.
The sale of the Sir Aston Webb-designed landmark Grade I building, which stands on the south west corner of Trafalgar Square and provides an entrance to the Mall leading to Buckingham Palace, is part of a government initiative to sell underused official buildings.
It will be transformed into a landmark hotel, with the building remaining in the ultimate control of taxpayers, as the Government has retained the freehold of the property.
To restore the Grade-I listed building, PIC is working closely with English Heritage and Westminster City Council.
Serrano said: "We will deliver the very finest hotel for London, with an unrivalled history and location."