"Doctors have concluded the tests, and these have revealed a recurrence of a previous lung infection, for which Madiba is receiving appropriate treatment and he is responding to the treatment," the presidency said on Tuesday.
"President [Jacob] Zuma thanks the public for continuous support to former president Mandela and his family at this time."
Eight soldiers were screening vehicles entering 1 Military Hospital, on the outskirts of Pretoria, on Tuesday morning after the former president spent a third night there.
Military police stopped and inspected numerous cars at the hospital's entrance before allowing them onto the premises.
Vehicles belonging to news crews, including an outside broadcast vehicle, were turned back at the entrance.
Several journalists milled outside, metres away from the security checkpoint.
'Doing very, very well'
On Monday, Defence Minister Nosiviwe Mapisa-Nqakula said Mandela was "doing very, very well" while undergoing unspecified medical tests at the military hospital.
She offered the first government confirmation that Mandela (94) was at the hospital. He has received military medical care since 2011.
The office of President Jacob Zuma also confirmed on Monday that Mandela was fine, but did not offer further details.
"Mandela had a good night's rest," presidential spokesman Mac Maharaj said in that statement. "The doctors will still conduct further tests today. He is in good hands," he said. On Saturday, Zuma's office announced that Mandela had been admitted to a hospital in Pretoria for medical tests and care that was "consistent for his age." Zuma visited Mandela on Sunday morning and found him to be "comfortable and in good care".
In February, Mandela spent a night in a hospital for a minor diagnostic surgery to determine the cause of an abdominal complaint. In January 2011, Mandela was admitted to a Johannesburg hospital for what officials initially described as tests, but what turned out to be an acute respiratory infection.
Other health problems
Mandela has had other health problems. He contracted tuberculosis during his years in prison and had surgery for an enlarged prostate gland in 1985.
In 2001, Mandela underwent seven weeks of radiation therapy for prostate cancer, ultimately beating the disease.
Mandela, who spent 27 years in prison for fighting racist white rule, became South Africa's first black president in 1994 and served one five-year term.
The Nobel laureate later retired from public life to live in the remote village of Qunu, in the Eastern Cape, and last made a public appearance when the country hosted the World Cup soccer tournament in 2010.
He has grown increasingly frail in recent years. – Sapa