Almost parallel to Old Court House Street, there is a stretch of road, known as Council House Street that connects the southwest portion of the Government House with the southwest part of the Dalhousie Square (BBD Bag). However, today there is no building on the stretch called the Council House, as it was demolished long back, in 1800.

After the Battle of Plassey, as Calcutta started to expand rapidly, the British East India Company felt the urgent need to find a suitable building, which can be used as the Council House, as there was no Council Room for a period of twelve months to carry out the necessary business of the settlement. In 1758, the dwelling house of the late Richard Court was purchased for the purpose, along with an adjacent building for the temporary residence of the Governor. The building of the Council House was probably a house near the old hospital, which once situated very near to the St. John's Church and is now known as Garstin Place. The Council House remained there in use till 1764, when a new building was built for the purpose, on the Esplanade and gave its name to the street.

Council House Street, which starts from the southwest portion of Government House and ends at the southwest part of BBD Bagh, former Tank Square, earned its name from the old Council House, which once stood on the western portion of Government House and the Fort William College was located at the corner of Council House Street. However, the building was unceremoniously pulled down in 1800, to make room for the present Government House.

It is interesting to note that, the Council House, which once stood on the present Council House Street, was one of the several buildings demolished to make room for the majestic Government House. Today, the present South-West wing of the Governor’s House, containing the Governor’s private apartments, stands on part of its site.

The Council House, which is lost forever, was intended to house a large number of Government offices, as well as the Council Chamber. However, for the first ten years of its existence, it was appropriated by the Governor as his official residence until in 1775, when Warren Hasting relinquished it for the convenience of public business and hired Buckingham House to its East. However, as Lord Wellesley found it inconvenient and derogatory, he took the decision to construct a dignified building and that resulted the demolition the building of the Council House, which once stood on the western portion of Government House. Today, the Council House Street exists, without the existence of the Council House.