The country has now recorded more than 230 cases of the virus.
The news of the death comes after people in a crowded camp for the displaced close to the capital, Juba, had tested positive.
The International Committee of the Red Cross (ICRC) said it will be very hard to control the spread of the disease because 30,000 people live crammed together, with three people sharing a mosquito net.
Extra hygiene measures are in place but it is not possible to keep people at a safe distance from each other.
The ICRC said it was also worried about Covid-19 spreading in overcrowded prisons and hospitals.
More than four million people have been displaced by conflict in South Sudan.