The City Hall called "Das Neue Rathaus" (The New City Hall) was built from 1894 and 1900, first plans were from 1880. The inauguration was on the 27th of december 1900.
The tower is 61m high.
In the Second World War the City Hall was not heavily damaged comparing to rest of the city where about 90 % of the buildings were destroyed.
So the City Hall is still the original building.
For more information look at wikipedia (only in German):
History of Brunswick
History of the City Hall
English (shorter) version:
Braunschweig