But in neighbouring Bulgaria, the government was refusing to bow to growing demands for fresh elections, and denounced opposition attempts to force them from office following the worst scenes of violence in the Balkan country since the overthrow of communism in 1989.
Mr Milosevic's remarkable climbdown was announced during a meeting of senior government officials and the Belgrade University students who have formed the backbone of the protests. "The Serbian government representatives agreed with the necessity that the will of the citizens expressed at local elections must be fully respected," a statement signed by Serbia's two deputy prime ministers, Ratko Markovic and Nedeljko Sipovac, and a delegation of students said. The statement also promised to punish those responsible for "forging" the elections - code for an imminent government reshuffle and a purge of hardliners in Mr Milosevic's Socialist Party.
"The talks were not unpleasant, they immediately expressed readiness to accept our demands," Cedomir Jovanovic, a student protest leader, said. Opposition leaders and foreign diplomats were confident Mr Milosevic would abide by the findings of an international report accusing the Socialists of trying to steal the elections in 14 towns and cities, including Belgrade.
It is still possible that he might try to award his party more seats in city councils than it actually won, but the opposition Democratic Party said it would gladly take power first and worry about other problems later. "We are open to any dialogue. We want to introduce reforms in a gradual way," said Miroljub Labus, one of the Democratic Party's vice-presidents. Other rumoured concessions included an invitation to the opposition to form a grand coalition, pending new Serbian elections later this year. There was even talk of secret contacts between Mr Milosevic and the White House about his possible resignation, to be followed by a three-month transition period in which the presidency would be taken by Milan Panic, the California businessman who was Yugoslav Prime Minister in 1992. The outgoing mayor of Belgrade, Nebojsa Covic, is tipped to become Prime Minister in place of the much-despised Mirko Marjanovic.
While the Serbian opposition cautiously began celebrating, their counterparts in Bulgaria, who had to some extent taken their lead from the Serbian protests, returned to the streets. At an emergency meeting convened by President Zhelyu Zhelev, the governing Socialists ruled out general elections demanded by opposition protesters.
"The government of Bulgaria clearly states that violations of law and social order as well as immediate parliamentary elections at this moment are destructive for the country," a government statement said. The previous night many of the party's MPs had to be rescued by riot police from a crowd of more than 50,000 which had blocked their exit from the parliament building in Sofia.
Although Serbia and Bulgaria are ruled by largely unreconstructed ex- communists, the Bulgarian protesters are trying to oust a legitimately elected government which has two years of its mandate to run. "In Belgrade the protest is against a form of dictatorship, but here it is basically about economic mismanagement," said Georgi Apostolov, deputy editor of the pro-opposition Kontinent newspaper. And while the protests in Serbia have largely been peaceful, those in Bulgaria turned violent after just five days.
Serbia's opposition, page 13