Clydesdale Bank International, which provides savings products for expatriates, is no longer taking deposits. Staff have been told to manage the run-down of the existing customer book, which will take it until next year.
Andrew Pearce, retail director at Clydesdale Bank, said: "We have been reviewing our strategy across the whole of Clydesdale Bank and are concentrating our efforts on what we do best in those areas we are strongest.
"Our focus now is to work with colleagues and customers to ensure that everybody understands this change, and that we manage a smooth transition."
Clydesdale Bank, which is owned by National Australia Bank, is retrenching to its heartlands in Scotland and northern England after being hit by bad debts caused by the commercial property price crash.
The Glasgow-based bank said there will be no immediate impact on existing customers as a result of its decision but said customers will be allowed immediate access to their funds, regardless of any product restrictions.
The 19 people employed in the business will remain as they manage its back book of business.
A spokesman said: "Ultimately something will happen to them. At this stage, whether there are redundancies, we don't know."
Clydesdale Bank International's managing director James Blower left at the beginning of the month but the spokesman said Mr Blower had not known about plans to shut the business when he exited.
Mr Blower has been replaced by Peter Symes from Yorkshire Guernsey, owned by Yorkshire Building Society, where he was managing director.
Clydesdale Bank International was established in 2008.