Cases increased to 24 by last night while a further 27 people are suspected to have the illness, Nicola Sturgeon told the Scottish Parliament.
Of the total 51, 14 are being treated in intensive care.
The single death from the outbreak was named locally as Robert Air, 56, from the Seafield area of Edinburgh.
Ms Sturgeon said: "It remains the case that there is no identified link between these cases other than an association with the affected areas in the south west of Edinburgh."
Two people who have contracted Legionnaires' have been discharged from hospital.
The other patients are being treated for the disease either in general hospital wards or in the community.
Ms Sturgeon passed on her condolences to the family of the dead man.
She said the outbreak of the disease in the capital was the worst the country has seen since the 1980s.
She told MSPs at Holyrood: "In Scotland, we would normally expect to see around 30 to 40 legionella cases each year. Typically around half of these cases are contracted abroad, but we also see indigenous cases, and it is not unusual to see single sporadic cases of community-acquired legionella.
"Across Europe, outbreaks are not uncommon, with dozens of outbreaks per annum and thousands of cases.
"However, 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."
She explained that Legionnaires' disease was an "uncommon but serious form of pneumonia, caused by bacteria that are distributed widely in both natural and artificial water supplies".
Ms Sturgeon said: "In most cases, the disease is caused by the inhalation of water containing the bacteria and common sources can be showers, air conditioning, cooling towers, or humidifiers."
Ms Sturgeon said the increase in cases was "not unexpected" and warned the number of people affected could continue to rise for the next week.
The Health Secretary said: "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 five or six days.
"It is therefore vital that we ensure that the public are aware of the symptoms, and that the NHS is primed to quickly identify, diagnose and treat cases."
She said she was "satisfied" appropriate action had been taken to alert the public and prepare medical services, adding: "It is encouraging that a number of the existing cases are responding well to treatment."
She also said staff at NHS Lothian had acted "swiftly" to set up a multi-agency incident management team to deal with the outbreak.
That was done on Sunday, three days after the first case of the disease was confirmed.
After the initial case came to light a week ago, a second was confirmed on Saturday, with another two on the Sunday.
At that time it was judged the source of the disease was unlikely to be an indoor one, such as a spa or swimming pool.
Most outdoor outbreaks of Legionnaires' are linked to cooling towers, and Ms Sturgeon said action was taken on the Sunday night and Monday to treat these.
Cooling towers at four sites in the area were "shock dosed" with chemicals to treat bacterial growth, Ms Sturgeon said.
But she added that she had been told last night that some of the towers may need further treatment.
The Health Secretary said discussions were taking place between the City of Edinburgh Council and the Health and Safety Executive to determine what, if any, further treatment is considered appropriate.
She also said that over the last two days a further three towers in two different sites had also been tested and treated.