The elderly patient at Copenhagen's Herlev Hospital had already been admitted suffering from other illnesses, according to a report by Danish state broadcaster DR.
The death is the second to take place at the hospital, following the death on Saturday of an 81-year-old patient who had also been admitted for other serious illnesses.
On Thursday night, an 80 year old in North Jutland was found to have been infected with the coronavirus after dying from a heart attack. Authorities have not yet officially recorded that death as due to coronavirus.
Denmark's national medical laboratory Statums Serums Institut on Saturday night said it expected to record the two first deaths in its coronavirus statistics.
The Danish Health Authority (Sundstyrelesen) reported 79 new cases of infection on at 10am on Sunday morning, taking the total number known to be infected in Denmark to 864.
The actual number of infected people in Denmark is estimated to be higher but the Danish authorities are now only testing the most serious cases. Those with mild symptoms are asked to stay at home and not infect others.
According to the National Board of Health, up to 580,000 Danes risk being infected with the coronavirus and between 1,680 and 5,600 Danes risk losing their lives, with the number of deaths expected to be at the lower end of the scale.