Such an event was feared by the Health Ministry and the Prime Minister's Office, which launched an expensive vaccination campaign just a few days ago.
The death "apparently" had nothing to do with the shot, the ministry said.
The man, who lived in Bat Yam but was not further identified, had been undergoing ambulatory dialysis for acute renal insufficiency and also suffered from heart disease. His family refused to allow an autopsy, the ministry said, so the exact cause of death cannot be determined.
The ministry said on Sunday that the first group to get their shots are those at high risk for complications of the H1N1 flu and that it is "likely" that his serious diseases caused his death.
By instruction of Prime Minister Binyamin Netanyahu, some 7.3 million doses of H1N1 flu vaccine - enough for every resident over the age of six months - have been ordered at a cost of NIS 500 million.
Although Health Ministry public health officials advised Netanyahu - who holds the health portfolio, though the ministry is run by Deputy Health Minister Ya'acov Litzman - to import many fewer doses, Netanyahu decided otherwise.
The prime minister appeared at the press conference that launched the campaign and called on everyone to get the shot as a "civic duty" even though he himself did not bare his arm; his spokesman said Netanyahu would "consult with his personal physician" before deciding whether to be vaccinated.
Litzman, who was also at the press conference, didn't take the shot, either, but Health Ministry director-general Dr. Eitan Hai-Am did, as the cameras clicked.
The ministry said that the H1N1 vaccines are strongly endorsed by the World Health Organization.
In the fall of 2006, after two elderly men died a few days after getting vaccinated at their health fund clinic against ordinary flu, Israelis nearly stopped getting the shots. Even then-health minister Ya'acov Ben-Yizri, 79, getting a flu shot on camera to build public confidence flopped. As a result, many hundreds of thousands of doses had to be destroyed because they are effective for only a year.
The H1N1 vaccine may also be unusable next fall and winter if the virus mutates into a different strain.
Several months ago, a ministry poll showed that at least a quarter of the public said they were not interested in getting an H1N1 flu shot. So far, the queues for getting the H1N1 flu shot have been short.
Since Rosh Hashana, more than 13 percent of the public has gone for a seasonal flu shot, including only about half of the elderly who are at high risk.
Dr. Itamar Grotto, the Health Ministry's chief of public health, said on Sunday night that some 10,000 people - especially medical staffers, as well as patients at high risk for complications of the H1N1 flu - have been vaccinated so far.
After any vaccination, the individual must wait in the clinic for 20 minutes to make sure he is not among the very rare people who have a severe reaction from an allergy to the egg white in which the vaccine is prepared.