Another 19 people are suspected of having the illness in the Lothian health board area but no more fatalities have occurred since yesterday when one man died, Nicola Sturgeon said.
Twelve of the 21 patients are in intensive care, two have been discharged and the others are either being treated in hospital or in the community.
The number of confirmed cases rose by four from the 17 announced at a previous update this morning. There are also four more suspected cases.
NHS Lothian said the patient who died was in his 50s and had other health conditions. He was being treated at Edinburgh Royal Infirmary.
The majority of confirmed cases are linked geographically to the Dalry, Gorgie and Saughton areas of south-west Edinburgh
This morning 15 people were being treated in intensive care but this has come down to 12.
The number of cases is expected to continue rising until the weekend because it can take as long as two weeks for symptoms to show.
Legionella bacteria is commonly found in sources of water, such as rivers and lakes, NHS Lothian said. The bacteria can end up in artificial water supply systems such as air conditioning, water services and cooling towers.
Legionnaires' disease is contracted by breathing in small droplets of contaminated water. Symptoms are mild headaches, muscle pain, fever, persistent cough and sometimes vomiting and diarrhoea. They can begin at any time between two and 14 days after being infected.
The first case in the current outbreak was identified on Thursday May 31.
About half of those who catch the disease will also experience changes to their mental state, such as confusion.
Legionnaires' disease is not contagious and cannot be spread directly from person to person.
Ms Sturgeon said officials expected to see more cases as the week goes on.
"Given the incubation period for Legionnaires' disease, we would expect to see further cases emerge over the next few days," she told a press conference in Edinburgh.
The process of gathering information from the confirmed and suspected cases was continuing, she said.
"As you would appreciate, that's a complex process because it involves, in many cases, dealing with critically ill patients.
"NHS Lothian have brought additional public health staff to bear to make sure that as much information about the behaviours, patterns and the recent histories of these patients is being gathered.
"What we can say is that no link has been identified between these patients other than a general association with the affected area in the south-west of Edinburgh. What that does is underline the view that the source of this infection is an outdoor community source and not an indoor-specific source, such as would be the case if it was a spa in a hotel.
"That points, as we have been saying, to cooling towers in the south-west of Edinburgh."
Work around the towers is being led by the city council and the Health and Safety Executive.
Ms Sturgeon said: "Samples have been taken from all of those towers and all of them have been subject to what is called shock treatment which is effectively chemical treatment to deal with the risk of ongoing infection, and there will be sampling over the course of today around these towers."
The process of identifying the infection source was complicated, she said.
"I'm advised by experts that very often with Legionnaires' disease, it's not possible to identify a particular source beyond reasonable doubt so often what is dealt with is a balance of probabilities of where the source is likely to be.
"The process of identifying the source is a complicated one and there are three main strands to that.
"Firstly, epidemiological which is looking at the pattern of cases to try to narrow down to as small an area of possible infection as possible.
"Secondly, microbiological. That's about taking samples, looking at whether Legionella is in existence in a tower and whether it matches a particular strain of Legionnaires' that the patients have.
"Thirdly, it involves the Health and Safety Executive inspection process, looking at the management process around these towers and, if necessary and appropriate, carrying out physical inspections of the towers. And aspects of those processes are ongoing."
Ms Sturgeon said: "This is the most significant Legionnaires' outbreak we have had in Scotland for a long, long time, perhaps since the early 1980s.
"It's a significant outbreak and it's the cause of understandable concern, but it's also resulting in all of the relevant agencies working together very closely, both to manage the outbreak and ensure that patients have the right treatment, and to identify the source as quickly as possible.
"There are in any given year in Scotland some 35-40 cases of Legionnaires' disease identified. About half of them are thought to be contracted overseas by people who have been travelling."
The Scottish Health Secretary activated the Scottish Government Resilience Room (SGoRR) last night.
She added: "I should say at this stage that it is not routine practice for SGoRR to be activated as a result of a public health outbreak within one health board area.
"The normal course of dealing with that would be as did happen in NHS Lothian through the incident management team.
"What triggers SGoRR to be activated is if the scale or the complexity of an outbreak leads to the conclusion that it`s not appropriate for the response to be led simply by a health board in one area."
The significant spike in cases over the course of yesterday raised issues about NHS Lothian's capacity to deal with the cases and the conditions were right for the activation of the resilience room, she said.
The message to the public is that for the general population, the risk is low.
Ms Sturgeon added: "As with many conditions, those with underlying health conditions, those in particular risk categories, are at higher risk.
"It's also important to stress that Legionnaires' disease is not contagious. It cannot be passed from person to person. It cannot be contracted from drinking water.
"The other message to the public, of course, is anyone who has concerns about symptoms - and the general symptoms of Legionnaires' disease would be headache, a cough, fever, muscle pain - anybody with symptoms that are causing concern is advised to contact their GP or NHS 24.
"NHS 24 has established a special advice line for those concerned about Legionnaires' disease, which has just gone live. The number for that is 08000 858 531.
"NHS Lothian has prepared an information leaflet which will, over the course of today and into tomorrow, be sent electronically to all community contacts in the area and door to door to give people the advice and the reassurance that they need.
"The resilience committee will meet again this evening to update on progress around all of the things that I've been talking about.
"I will update (the Scottish) Parliament in a statement tomorrow morning and we will keep you all as updated as we possibly can in all aspects of information as we have it."
The belief that a cooling tower is the likely source of the outbreak comes from a number of factors, Ms Sturgeon said.
The scale of the outbreak suggested this, together with the fact the victims are not linked, other than the area they are in.
"The only link we know, so far, is that all of the affected patients have got links to this particular area where the cooling towers are. Other possible sources are not being ruled out."
The towers are the "probable sources of infection".
Ms Sturgeon added: "I think it would be right and proper to say that the minds of any of the experts working on this are not closed to other possible sources of infection."
The outbreak of Legionnaires' disease is a significant event for NHS Lothian, although NHS boards have plans for outbreaks such as this, she said.
"This is an ongoing situation. I very much hope we don't see anybody else lose their life through this outbreak. Obviously it's an ongoing outbreak: the number of cases are continuing to increase. I expect them to increase further.
"To some extent, how many cases we will see over the next period and for how long will depend on whether this was a single incident of a cooling tower emitting a contaminated cloud or whether it was something that was ongoing for a few days until we did the shock treatment.
"As (the Scottish) Health Secretary, working with my colleagues in NHS Lothian, our principal concern is managing this outbreak from a public health perspective and making sure the right things are done to bring this outbreak to an end. And that's what we're very focused on."
The holiday weekend had "absolutely not" slowed the response to the outbreak, she insisted.
"All of the things that should have been done have been done, and have been done in the right order. In my estimation, that has not had any bearing on this whatsoever."
Some of those infected were at last month's Scottish Cup Final between Hearts and their Edinburgh rivals Hibernian, with some also at Hearts victory celebrations. Hearts FC's Tynecastle ground is in the affected area.
Ms Sturgeon said: "I don't think it's surprising, given the area we're talking about, that some of them would have been at the football."
Dr Duncan McCormick, consultant in public health medicine and chairman of the incident management team at NHS Lothian, said those suffering from the disease are aged from mid 30s to late 80s with the majority at the upper end of the age range. The majority of cases are of men.
The antibiotic used to treat those infected was said by medics to be "very effective", while two seriously ill patients have now been discharged from hospital, which are "encouraging signs".
Dr McCormick added: "We are confident we have identified the source and we have taken steps to rectify that. So while we expect to see more cases over the next few days, we would expect after a few days, maybe five or six days, to start to see a decline."