South Africa's President Cyril Ramaphosa has warned that the coronavirus will turn into a "national crisis" and its impact will be "huge".
His warning came after the first case of the coronavirus was detected in South Africa.
The patient is a 38-year-old man who visited Italy with his wife.
This brings to 27 the number of coronavirus cases reported in Africa. Algeria is worst-affected, with 17 cases, 16 of them in the same family.
The South African couple, who have two children, were part of a group of 10 who returned from Italy on 1 March, Health Minister Zweli Mkhize said.
The man and the doctor who first treated him were both in self-isolation in the eastern KwaZulu-Natal province, he added.
A tracer team had been sent to KwaZulu-Natal to identify people who might have been in contact with the man and the doctor, Dr Mkhize said.
Mr Ramaphosa urged South Africans not to panic and to get immediate medical help if they showed symptoms of Covid-19.
"It will have a huge impact on a number of things… travel… our economy. It is already showing signs of a negative impact on tourism.
"The effect will be big. South Africans will need to be prepared," he was quoted by privately owned Eyewitness News as saying.
For now, South Africa is portraying its detection of the country's first Covid-19 patient as a sign of success – as proof that the public health systems of the continent's most-developed and sophisticated economy are working as planned.
"This is not… a failure. Our health systems [are] able to detect and rapidly identify cases," the National Institute for Communicable Diseases tweeted within hours of the announcement of the infection in KwaZulu-Natal.
The continent as a whole has clearly benefitted from the remarkably slow arrival of the virus in Africa – a window of several weeks that has enabled the World Health Organization (WHO), national governments and other groups to build up testing and treatment capabilities.
But in South Africa, there are particular concerns related to the country's long-running HIV epidemic – the worst in the world.
More than seven million people here are living with a virus that seriously weakens immune systems, leaving many people potentially more vulnerable to Covid-19.
Elsewhere in Africa, some people have taken comfort from the fact that many governments have recently been forced to confront the dangers of successive Ebola outbreaks – an experience that has strengthened the capacity of many health systems to detect and respond to a far deadlier virus.
But there are also deep concerns that – on a continent still wrestling with entrenched poverty, plagued by conflicts, and poorly served by over-stretched health services – a serious outbreak of this new coronavirus could quickly overwhelm some countries.
Two other South Africans who were admitted to hospital in Japan with the coronavirus have recovered, and would return home soon, other officials said.
The pair had been hospitalised in Japan after being taken ashore from the Diamond Princess cruise ship they were working on.
People aboard the ship were first quarantined in January when a passenger who disembarked in Hong Kong tested positive for the coronavirus.