Wye Valley NHS Trust, which runs Leominster Community Hospital, says anyone who has had symptoms of diarrhoea and/or vomiting in the previous 48 hours should not visit.
A spokesman for the trust said: "Wards at Leominster Community Hospital have been closed to visitors following confirmed cases of norovirus. Please only come to the hospital if you really need to. Anyone who has symptoms of diarrhoea and/or vomiting in the previous 48 hours should not visit."
There are two wards at Leominster Community hospital, Minster and Eaton. These are split into smaller rooms of single or five bedded bays, all single gender.
The NHS describes norovirus, also called the "winter vomiting bug", as a stomach bug that causes vomiting and diarrhoea. It can be very unpleasant, but usually goes away in about two days.
The main symptoms of norovirus are feeling sick (nausea); diarrhoea and being sick (vomiting).
People may also have a high temperature; a headache; aching arms and legs.
The symptoms start suddenly within one or two days of being infected.
The NHS says you can usually treat yourself or your child at home. They say the most important thing is to rest and have lots of fluids to avoid dehydration. People "will usually start to feel better in two to three days."