JOHANNESBURG — Nelson Mandela was seriously ill but has steadily improved over the last few days after being diagnosed with a lung infection and undergoing gallstone surgery, President Jacob Zuma said Thursday.

It was the first official acknowledgment that Mandela’s condition had been grave, and came 13 days after the anti-apartheid icon was brought to a hospital in the capital, Pretoria. The government initially said Mandela, 94, was undergoing medical tests. The information that followed was terse and sometimes contradictory.