The agency is understood to have beaten Instinctif Partners and FTI Consulting in a final three-way pitch for the account, which had been handled by Citigate Dewe Rogerson for five years.
The LSE Group, which also owns assets including the Italian Stock Exchange, has given the agency an international remit with a focus on corporate and financial PR, as well as strategic work.
The appointment comes ahead of the group announcing its preliminary results for the year ending 31 March, and as London sees a major boost in IPO activity as economic prospects brighten.
This spring the exchange reported a record 91 per cent jump in the amount of capital raised, with recent major flotations including Bonmarche and Poundland.
Finsbury declined to comment but it is thought its team will be led by founder Roland Rudd and partners Guy Lamming and David Henderson.
A spokeswoman for The LSE Group confirmed the appointment but declined to comment further.