Further deaths from an outbreak of Legionnaires' disease cannot be ruled out, doctors warned, as people from further afield fell ill.
There are 24 confirmed cases and 37 suspected cases of the infection, with 12 people being treated in intensive care.
Chairman of the incident management team, Dr Duncan McCormick, of NHS Lothian, said: "We're hoping to see no further deaths.
"We're doing everything we can in terms of early diagnosis, appropriate treatment and intensive care, but I think we can't rule out any further deaths at this stage."
Two patients are being treated outside of the NHS Lothian area - in the Highlands and in the North of England.
Scottish health secretary Nicola Sturgeon said: "I want to stress that, although these patients are being treated elsewhere, they are considered part of the south-west Edinburgh outbreak. They have had association with the affected area."
Ms Sturgeon updated the figures at a press conference in St Andrew's House, the Scottish government ministerial headquarters in Edinburgh.
Those going for medical treatment now are generally not as ill as at the start of the outbreak, she said.
"People that are being admitted to hospital with the symptoms of Legionnaires' disease are generally at the moment less ill than had been the case earlier in the outbreak.
"We're seeing fewer people, by no means nobody, but fewer people who are admitted to hospital going into intensive care and we're seeing an increase in the numbers of people who are actually well enough to be treated in the community.
"Clearly, the numbers of confirmed and possible cases have continued to rise over the course of yesterday. This is not unexpected and we expect to see further cases over the next week.
"The incubation period for Legionnaires' disease can be up to two weeks, although it will more normally be in the region of five or six days."
Although the source has not been identified, a fresh round of chemical treatment is under way at cooling towers in the industrial area of the capital where the outbreak is centred.
The actual source may never be conclusively identified, the Health Secretary said.
"I would hope that over the next few days those investigations will start to deliver more specific answers on where the source of the outbreak might be."
NHS Lothian has begun delivering leaflets to households in the affected area, designed to answer questions about the disease and provide helpline numbers.
Ms Sturgeon said: "I think there are ultimately 35,000 going to be delivered. The first 15,000 are already in the process of being distributed but it's going to go to every door.
"I'm not sure if it's the first time NHS Lothian has ever done a leaflet drop like this but, whether it is or not, it's the right thing to do because information is key to make sure people are alert, aware of the symptoms and that they know what to do."
A newly set-up helpline has received dozens of calls already, she said.
Earlier in the Scottish Parliament, Ms Sturgeon passed on her condolences to the family of the dead man.
The health secretary said Scotland would normally deal with around 30 to 40 Legionella cases each year, with around half of the cases contracted abroad.
She said: "Outbreaks of the size we are currently seeing here in Edinburgh are rare in Scotland. The last time we had an outbreak of this scale was, I understand, in the 1980s in Glasgow."
Legionnaires' disease is an uncommon but serious form of pneumonia, caused by bacteria distributed widely in natural and artificial water supplies.
The health secretary said she was "satisfied" that appropriate action is being taken to alert the public and prepare medical services.
NHS Lothian staff acted "swiftly" to set up a team to deal with the outbreak, she said. That was done on Sunday, three days after the first case of the disease was confirmed.