Mandela, 94, had been ill for some days before being transferred in the early hours of Saturday to a hospital in Pretoria.
It is the third time this year he has been admitted to hospital.  The BBC's Karen Allen in Pretoria says there is a sense of calm across South Africa, and also a quiet hope that the man who led the fight against apartheid may regain his strength once again.
Mandela's wife, Graca Machel, cancelled a scheduled appearance in London to remain at her husband's bedside.
On Saturday, presidency spokesman Mac Maharaj said that although Mandela was again suffering from pneumonia, he was breathing on his own - "a positive sign".
Nelson Mandela's medical condition "serious" - spokesman
South Africa’s former president Nelson Mandela's medical condition was "serious this time", presidential spokesperson Mac Maharaj says.
The 94-year-old anti-apartheid hero and the country's first black president was taken to hospital early on Saturday for a recurring lung infection.
"The situation is serious this time but doctors have assured us he is comfortable," Maharaj told television station eNCA.
Mandela's wife at his bedside after cancelling London trip
Nelson Mandela's wife Graca Machel called off a trip to London this week to be with her ailing husband who was readmitted to hospital with a lung infection on Saturday.
"Mrs Graca Machel cancelled her appointment in London on Thursday," South African presidency spokesman Mac Maharaj said.
"This morning when Madiba was hospitalised she accompanied him and she is with him at the hospital," he said, using Mandela's clan name.
Machel was scheduled to attend a summit on hunger in London hosted by British Prime Minister David Cameron which opened on Saturday.
South Africa's revered first black president and Nobel Peace Prize winner is in a "serious but stable" condition in a Pretoria hospital, his fourth hospital stay in seven months.
Mandela hospitalized with lung infection
Former South African president Nelson Mandela, 94, was readmitted to hospital on Saturday with a renewed lung infection and is in "serious but stable" condition, the presidency said.
"This morning at about 1:30 am (2330 GMT Friday) his condition deteriorated and he was transferred to a Pretoria hospital. He remains in a serious but stable condition," President Jacob Zuma's office said in a statement.
It marks the second hospitalisation for the ailing anti-apartheid hero in two months. On April 6 he was released from hospital after being treated for pneumonia during a 10-day stay.
Nelson Mandela 'in good shape and in good spirits'
Nelson Mandela has been visited at home by ANC leaders who said later he was "in good shape and in good spirits". The first pictures of the former South African president since he left hospital show him surrounded by well-wishers at his home in Johannesburg.
The 94-year-old was discharged on 6 April after more than a week of treatment for pneumonia.
It was the third time in the past four months that the Nobel Peace Prize laureate had stayed in hospital.
President Jacob Zuma and ANC Deputy President Cyril Ramaphosa were among senior ANC officials who called in on Mandela at his home in Houghton and were given an update by his medical team.
"After receiving a briefing from the medical team, the national officials are satisfied that President Mandela is in good health and is receiving the very best medical care," the ANC said in a statement.
President Zuma said Mandela "shook hands and even smiled