The theft led to BT business customers and home users in the area being cut off from their phone and broadband internet services.
The exchange is located at Farm Street, off Park Lane, and the thieves are said to have entered the building this Wednesday evening between 8pm and 9pm.
They ripped out servers, routers and network cards, which can all fetch a high price on the black market.
Last December, thieves raided the Verizon Business datacentre in London's Kings Cross. In that raid, a number of City firms were affected.
It is not known at this stage whether the spate of thefts are linked.