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He was ordered to step down by the Yugolsav President, Slobodan Milosevic, after marathon talks with the United States special envoy, Richard Holbrooke.
Mr Karadzic is currently a fugitive after an arrest warrant was issued for him last week by the International War Crimes Tribunal.
He was indicted last year, along with his army chief, Ratko Mladic, for crimes of genocide and other atrocities.
His continued refusal to resign was jeopardising the first post-war elections in Bosnia, due to be held on 14 September.
Under the Dayton peace accords, which brought an end to the war in the former Yugoslavia last year, indicted war criminals cannot stand for public office.
Mr Holbrooke announced the agreement in Belgrade, saying, "He [Karadzic] will not appear in public, or on radio or television or other media or participate in any way in the elections."
But he admitted he had failed in his aim to secure the extradition of Mr Karadzic.
"I want to make clear that we are not satisfied," he went on. "It falls short of our goals. Indicted war criminals should be at The Hague to face trial under due process of law, and that includes Mr Karadzic."
Mr Karadzic will be replaced by his deputy president, Biljana Plavsic, a Serb nationalist who is known to be opposed to the Dayton agreement.
The news was greeted cautiously by other international mediators, who have been struggling for six months to oust Mr Karadzic.
"We must await events since there have been so many agreements not properly implemented in this country," said Michael Steiner, deputy to the International High Representative in Bosnia.
Reaction from other political leaders in Bosnia was also lukewarm.
"I don't doubt that Karadzic will continue in his own way, creating a parallel system and obstructing the Dayton agreement and peace in the region, until he is completely removed," said the vice-president of the Muslim-Croat Federation, Ejup Ganic.
Radovan Karadzic and Ratko Mladic continued to evade arrest.
Nato troops carried out several raids on places he and Mladic were thought to have been hiding, but always without success.
The United States and European Union put pressure on the Bosnian Serb authorities to find them, and a £3.5m reward offered, again without result.
Mr Karadzic was finally arrested in Belgrade in July 2008 and is expected to face trial at The Hague, but General Mladic is still at large.
Slobodan Milosevic was forced to resign by his own people after he refused to recognise the victory of opposition leader Vojislav Kostunica in the 2000 general election.
He was jailed by the Serb authorities in April 2001, accused of corruption and abuse of power.
Two months later the Serbian government handed him over to the international community to stand trial on more than 60 charges of genocide and crimes against humanity at the International War Crimes Tribunal at The Hague.
His trial began in February 2002. Milosevic was found dead in his cell in March 2006. By the time of his death, the prosecution had completed its case but the defence was continuing, interrupted by Milosevic's frequent ill-health.
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