NHS Lothian said it is investigating the outbreak at the Western General Hospital after a “very small number of patients” died.
The health board did not confirm the number of fatalities, but said it was fewer than five.
A further six patients have tested positive for Covid-19, and the oncology ward has been closed to new patients.
Dr Donald Inverarity, consultant microbiologist, NHS Lothian, said: “Our thoughts are with the family of the deceased and I would like to express our sincere condolences.
“A multidisciplinary Incident Management Team was immediately established and all necessary infection control measures are in place. The situation will continue to be reviewed and monitored very closely.”
Current patients who would normally spend the weekend at home with family before returning to hospital on Monday are being asked to remain on the ward to reduce the risk of spreading the virus.
The health board’s Health Protection Team and the nationwide Test and Protect teams are carrying out contact tracing of visitors and outpatients, where necessary.
Routine Covid-19 screening of staff and patients is taking place as part of an enhanced regime.
Dr Inverarity said: “The enhanced outbreak testing regime allows us to identify staff and patients who have the virus before they become symptomatic.
“That gives us the opportunity to introduce enhanced infection control measures and barriers sooner than we would previously have been able.
“It allows staff to begin self-isolation and patients to be moved into cohorted groups before they become symptomatic, which in turn reduces the likelihood of passing on the infection to others.
“Patient safety is our main priority and while we understand that the request not to go home for the weekend may be upsetting, it is necessary.
“It will help reduce the risk of onwards transmission and protect their families and the wider community. I would like to thank patients and their families for their co-operation and understanding.”