Patients and staff at Furness General Hospital have had diarrhoea and vomiting.
Surgical and medical wards four, six and nine have been closed to admissions and transfers.
Visitors to affected wards are being advised not to visit with young children or if they have had symptoms within the previous 48 hours.
The trust has apologised for any disruption.
An outbreak of norovirus at a Cumbria hospital has forced the closure of three wards.