The AMO website has this description:
The former French Mission Building on Battery Path, Central, was built in 1917 by the French Society of Foreign Missions as its Procure of Hong Kong. The building was used briefly by the Government before Hong Kong fell to the Japanese in 1941. Immediately after the Second World War, it was made the temporary headquarters of the Provisional Hong Kong Government. The French Mission and the Hong Kong Govenmment signed a contract for the transaction of the building in 1952. Since 1953, the building was occupied successively by the Education Department, Victoria District Court, the Supreme Court and the Information Services Department. It housed the Hong Kong Court of Final Appeal between 1997 and 2015.
Dating from the Edwardian period, the three-storey building with a basement is constructed in granite and red bricks in the Neo-classical style. A chapel is incorporated in the north-west corner, where its cupola projects above the roof. It has undergone several alterations since the Second World War, but many historical architectural features can still be seen in the building. For example, the hall on the ground floor with its ornate columns, wooden staircase and vaulted ceiling and the well-proportioned inner courtyard qualify the building as one of the finest pieces of historical architecture in Hong Kong.
In the comments below, Herostratus notes the construction of this building was completed in 1919.