NHS Borders said Ward Seven at the hospital, in Melrose, is now closed to admissions after eight confirmed cases of the virus were identified among patients.
A small number of staff members are also currently symptomatic and awaiting test results.
NHS Borders said a Test and Protect team are in the process of tracing contacts, adding that all non-essential visits to the ward have been suspended.
Dr Ed James, Consultant Microbiologist, said: “A multidisciplinary team has been established to oversee the management of this outbreak and the situation will continue to be reviewed and monitored very closely.”
Despite the outbreak, the Scottish Borders region, which is currently in Level Two of the Government’s lockdown system, is due to be moved into Level One from Friday, December 11, at 6pm.