The $1.87 billion all cash transaction will see State Street take control of the correspondent banking, depository banking, fund administration and global custody parts of the Italian banking group's operations.
It will also acquire around $16 billion in cash deposits.
State Street's president and chief operating officer Jay Hooley said the deal would help the financial services company expand its customer base and build on its position in the fund accounting and offshore fund servicing markets.
It will also allow the company to move closer to achieving its aim of generating 50 per cent of its revenue from non-US operations.
Mr Hooley added: "We feel that our strong capital position allowed us to take advantage of this attractive market opportunity which will further enhance our international presence and add to our financial profile and capital-generation capabilities."
State Street expects that the acquisition, which it plans to finance using available capital, will increase its annual revenue by $427 million.
Earlier this month State Street announced it had agreed terms to acquire Mourant International Finance Administration as part of its plans to expand its global fund administration capabilities.
By Asim Shah