Mpshe told a packed news conference that was broadcast live nationwide: "An intolerable abuse of process has occurred which requires discontinuation of the prosecution."
Reacting to the decision on Tuesday, Zuma said: "There never was a case against me ... I have been vindicated. There is no cloud. There has never been a cloud... At the moment we have a country to run."
He had been accused of accepting bribes to thwart an investigation into wrongdoing by Thales, a French arms company. He has repeatedly denied the allegations.
A high court judge dismissed the charges of corruption, fraud, money-laundering and racketeering last year.
But the appeals court overturned that decision in January.
"An intolerable abuse of process has occurred which requires discontinuation of the prosecution"
Mokotedi Mpshe, acting director of South Africa's public prosecutions
Zuma, widely viewed as the probable next president of South Africa given the ANC's dominance, will now be able to assume office without a looming threat of a criminal trial.
A poll released on Friday suggested that the ANC would secure more than 64 per cent of the vote in the election, just short of the two-thirds majority that it currently holds, which allows the party to push through constitutional amendments.
Prosecutors have stressed that withdrawing charges has nothing to do with Zuma's guilt or innocence. They warned that opposition parties could file civil cases.
Adam Habib, a political analyst and vice-chancellor of the University of Johannesburg, said the court's decision to drop the charges did not prove that Zuma was innocent of corruption.
"If Zuma is interested in the integrity of his political administration, then he has to do something more," Habib said.
"He has to take the nation into his confidence, call a press conference and say exactly what has happened.
"If he does not do that, then his political administration will be compromised and the country will suffer irreparable damage."
Hundreds of Zuma's supporters celebrated in Johannesburg after the announcement.
"I'm very happy for the decision, hoping that this gives our president what he needs for us to go forward," Victress Iwabi, an ANC town councillor, said.
Earlier, about 20 supporters of the Congress of the People (Cope), which broke away from the ANC last year, had gathered outside the national prosecutors' office demanding that Zuma face trial.
"Comrade JZ must face his day in court. He has been tried in a secret office but the public deserves to know the merits of the case," Mgcini Tshwaku, the Cope youth secretary, said.
Zuma was deputy president for six years before he was sacked in 2005 by Thabo Mbeki, the former South African president, after being implicated in a trial that saw his former financial adviser convicted of fraud and corruption charges.
In May 2006, Zuma was acquitted of rape charges.