The branch in Church Street will close its doors on September 23.
Colleagues working at the branch will be offered alternative roles in the company.
It follows closures of branches in Madeley and Oakengates in recent years. Barclays still has a branch left in Telford town centre.
Barclays said the role of the physical branch is evolving, with fewer than 10 per cent of transactions now taking place inside a branch.
Only 20 regular customers use the Wellington branch exclusively for their banking, it added.
Everyday transactions can be completed at the Post Office with the closest located at 6 Bagley Drive, Wellington.
Julia Husband, a Barclays spokeswoman, said: “The decision to close a branch is never an easy one. However, customers are increasingly using alternatives to branches to do their banking. As a result, we are seeing a sustained fall in branch visits across the UK.
"This is reflected at the Wellington branch where there has been a 22 per cent reduction in cash counter transactions in the two years to March 2020. In addition, 85 per cent of our customers at the branch are also using different ways to bank.
"We will work with our customers and provide alternative options to ensure they can continue to manage their money and receive financial expertise when required.”