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Some 1,500 Serb troops overran the lightly-armed Dutch troops, despite two Nato air strikes on Serbian tanks inside the enclave.
The Bosnian Prime Minister, Haris Silajdzic, called the Nato response "too little, too late" and said the people of Srebrenica had been "betrayed".
Up to 30,000 refugees from the mainly Muslim town are now reported to be fleeing to the north and the United States has questioned the UN's ability to carry out its "humanitarian mission" in the region.
UN spokesman Alexander Ivanko old reporters: "It is our understanding that Srebrenica has fallen to the Bosnian army.
"Thousands of refugees are fleeing to a town called Potocari in the north of the enclave. The Dutch company that was in Srebrenica has also withdrawn to Potocari."
Thirty Dutch peacekeepers had been captured during the Serbian advance, Mr Ivanko added.
The 400-strong UN battalion had set up a "blocking position" south of Srebrenica in an attempt to defend the town.
Nato air strikes hit two Serbian tanks, but Dutch Defence Minister Joris Voorhoeve requested they were suspended after receiving "terror threats" from the Serbs.
Mr Voorhoeve said: "They threatened to shoot dead the 30 Dutch soldiers they are holding hostage and to raze the Dutch battalion's headquarters to the ground. And to raze Srebrenica to the ground."
But the Bosnian Serb Army commander, General Ratko Mladic, said in a letter to the UN that the offensive had been necessary to "neutralise terrorists" and accused the UN of failing to demilitarise the area.
And the general said that civilians and peacekeepers had no need to fear the Serb advance.
With Serb sights now set on the nearby Zepa safe area - according to a broadcast by a Serbian-controlled television station in Pale - US Defence Secretary William Perry said the now UN faced a serious challenge.
"This raises the question as to whether the UN force will be able to stay in Bosnia to perform the humanitarian mission," he said.
What happened in the weeks after the Serb offensive have since been described as the worst atrocity in Europe since World War II.
On 12 August 1995 the UN announced an investigation into reports up to 2,700 Muslim men had been shot dead with machine-guns and buried in mass graves. US aerial photographs appeared to show evidence of mass graves close to Srebrenica.
It is now thought that between 7,000 and 8,000 Muslim men were killed by Serb soldiers following the fall of Srebrenica.
Refugee accounts claimed General Ratko Mladic, the Bosnian Serb commander, addressed the Muslim captives and warned that 1,000 refugees would be killed for every Serb who died.
General Mladic and the Bosnian Serb President at the time, Radovan Karadzic, were both later indicted by the War Crimes Tribunal for genocide. 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.
One senior Bosnian Serb, General Radislav Krstic, was jailed for genocide for his part in the Srebrenica tragedy.
There have been a number of investigations into the massacre. In 2002 the entire Dutch government resigned after an inquiry blamed officials for giving the poorly-armed troops an impossible task to defend the enclave.
Initial findings of a commission involving the Bosnian Serbs admitted for the first time in June 2004 that Serbs had taken part in the killings.
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