A no-confidence motion is scheduled to be debated on Tuesday and requires the backing of a majority of 400 lawmakers to pass.
While there is mounting disgruntlement with the ANC over Zuma's leadership and his immersion in a succession of scandals, the party says it will resolve its leadership issues internally and won't allow its lawmakers to side with the opposition to bring down Zuma's administration.
A secret vote increases the chances of Zuma's ouster because members of the ruling party can vote him out without risking losing their jobs.
Opposition parties, who asked for the secret ballot, had threatened to challenge Mbete's decision in court after the Constitutional Court in June left it to her discretion to decide on the voting procedure.
The stalwarts wrote an open letter urging MPs to remember the pledge they made as Members of Parliament - to never conduct themselves in a manner that will diminish their integrity and that of the Republic of South Africa.
The party insisted that it did not oppose the vote being held in secret. "This is in the best interest of the country", Mbete said.
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The 32-year-old woman is now in a serious condition in hospital due to health problems which had been allowed to go untreated. Spotsylvania deputies made a house call on Saturday afternoon, where they said they found Moore with the three others.
But the success of the motion is likely to have harsh consequences for Mbete's own presidential ambitions as she will likely face isolation for her role in toppling Zuma and leaving the ANC vulnerable to the opposition's ridicule.
IFP Chief Whip Narend Singh said a successful vote of no confidence would be the nuclear bomb to cleanse the country of corruption, as African Christian Democratic Party (ACDP) leader Kenneth Meshoe expressed his "delight".
"(African National Congress) MPs now have no excuse", DA leader Musi Maimane said.
While some opposition parties have started celebrating the awarding of a secret ballot, their festivity may be short lived if the ANC manages to close ranks and the opposition fails to unite fully.
The matter was taken to court by the United Democratic Movement. "The people of South Africa also look to parliament for signals of hope", said Mbete, a longtime Zuma ally.
Parliament will vote on the motion on Tuesday and if it succeeds, Zuma, in power since 2009, and his entire cabinet would have to step down.
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