Airdrie Savings Bank, the last survivor of Britain's independent savings banks, is preparing for a "phased end to all business activities".
Staff were told the headquarters will shut, along with the remaining two branches in Bellshill and Coatbridge.
It is understood that 70 jobs are likely to be lost.
The bank is thought to have found the cost of new regulation to be too great for a small lender, and the Board of Trustees made the decision to close.
Chairman Jeremy Brettell said: "Whilst we are financially strong, a comprehensive strategic review of all future options concluded that we will not have - as a very small bank - the resources in the years ahead to provide the products and services our customers need in this increasingly digital world.
"The decision to implement a phased end of business activities is totally consistent with the over-riding responsibility of our Board of Trustees to protect the best interests of our customers, both now and in the future."
The closure came despite many of the new rules being introduced in response to the errors made before the financial crisis by Airdrie's much larger competitors.
The bank's chief executive Rod Ashley stressed the strategic review had "interrogated" all future options available to the bank.
He said: "Sad as the course of action outlined today is - in terms of our history and heritage - we are in absolutely no doubt it is in the best future interests of our customers."
Its profile was high when the Royal Bank of Scotland and Halifax Bank of Scotland had to be rescued. With a strong traditional ethos, it saw the number of customers increase.
In 2010, several of Scotland's most prominent business leaders - including Sir Brian Souter, Sir Tom Farmer and Ann Gloag - each put £1m investments into Airdrie Savings Bank, to help it expand.
However, in recent years, it has struggled as the industry has seen customers move their banking away from branches, and secure internet and mobile banking has required very high levels of investment.
Airdrie Savings Bank had already closed several branches in central Scotland. Baillieston, Muirhead, Motherwell and Shotts saw their branches close in August 2015.
The bank was founded in 1835, and ran out of a church and then two Airdrie shops, until its first branch was opened in 1883.
The arrangements for winding up a bank require it to offer depositors their money back. At Airdrie, the management is promising to help customers shift to other banks. The loan book is being handed over to TSB.
The regulator, the Prudential Regulation Authority, within the Bank of England, has to be informed of a bank winding up, but does not intervene where the directors or trustees are carrying this out in an orderly way.
Wendy Dunsmore of the Unite union, which represents staff, said: "This announcement is a bitter disappointment and will come as a shock to many.
"Unite will be doing everything we can to support our Airdrie Savings Bank members during this incredibly sad and difficult time, and to make sure their employment rights are protected.
"Our understanding is that the bank has had to meet extra costs due to increased regulation following the financial crash of 2008. So Airdrie Savings Bank has become yet another innocent victim of casino bankers."
She added: "Even at this late stage, we would urge the Scottish and UK governments and the banking regulators to work with us to explore all options for saving jobs.
"One of Unite's first steps will be to talk to the Scottish government's Financial Services Taskforce which was set up during the banking crash to help redundant workers back into employment."
Airdrie and Shotts MSP Alex Neil reacted angrily to the announcement.
He told BBC Scotland: "It really makes my blood boil that all these fat cats in the City of London who brought the financial crash upon us all these years ago, they're still walking away with big bonuses.
"The people who ran the Airdrie Savings Bank, many of them on a voluntary basis at board level, are now having to face this closure because of the regulation resulting from the greed of the fat cats in London."