This building in the Market Place is part of Brown's Estate Agency. It was empty for some time before that after being home to the Trustees Savings Bank. From 1907-47 it was Barclays Bank and before that Gurney and Company used it as their bank.
Other relevant information: The Museum evolved from a collection of artifacts assembled by the Mildenhall Natural History and Archaeological Society to mark the Festival of Britain in 1951 and displayed in two rooms on the upper floor of the old Town Hall in St Andrews' Street. The present premises are the Museum's third home. In 1968 the Museum moved to premises belonging to the Trustees Savings Bank( see the picture here) in the Market Place. When the lease ran out the Museum closed in 1982 whilst work was progressing on the conversion of two cottages and in 1983 the Museum opened in King Street opposite Bussens and Parkin.