Shaha Riza was a Middle East expert at the bank and Mr Wolfowitz's handling of the situation has been the subject of intense speculation as well as an investigation by a panel of directors.
Mr Kellems departs after advising Mr Wolfowitz at both the bank and the US department of defence during the invasion of Iraq, of which Mr Wolfowitz was a key architect.
The former aid said: "Given the current environment surrounding the leadership of the World Bank Group, it is very difficult to be effective in helping to advance the mission of the institution.
"I have tremendous respect and admiration for the bank staff and management."
The recent embarrassment surrounding Wolfowitz could not have come at a worse time for the bank, which is currently trying to spearhead an anti-corruption campaign.
A report containing the investigative committee's findings will be published shortly and Mr Wolfowitz will be allowed to respond to its findings.
So far, although under intense scrutiny and regularly questioned by the media about his standing at the bank, Mr Wolfowitz has refused to stand down.
US president George Bush has also leant his support to the former administration figure after Wolfowitz was nominated by the Bush administration to the World Bank post.
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