"This is the beginning of the end of Karadzic as a political leader, " he said.
Bildt's spokesman, Colum Murphy, said Saturday he believes Karadzic's reign will soon be up.
Earlier this year, two operations by Nato-led peacekeepers failed to catch Mr Karadzic in south-eastern Bosnia.
Several properties belonging to the Karadzic family were raided but their target was not found.
The two best known suspects are Bosnian Serb President Radovan Karadzic and his army commander, Mladic.
CNN: Bosnia talks founder as key Bosnian Serb fails to turn up
Mr Karadzic is verbally nimble, mentally sharp - and no doubt infuriating to many.
It helps that Slobodan Milosevic has a brutal-sounding name, and that Radovan Karadzic is memorably hairy.
Should America's willingness to use its courts this way worry anyone other than Mr Karadzic?
But even Mr Karadzic's arrest would not mean the downfall of the Pale clique.
April 23, 1995: The United Nations International Criminal Tribunal names Karadzic and Mladic as suspected war criminals.
Mr Karadzic's police can make it impossible to hold an election, at least in Bosnia's eastern half.
The sources said the speaker of the Serb parliament, Momcilo Krajisnik, has promised Bildt that Karadzic will resign.
Karadzic made no mention of stepping down as the president of the Serbs' self-declared state of Republika Srpska.
The Dayton peace process forbids Karadzic from holding any political office, even though he has remained in control.
July 21, 2008: Karadzic is arrested after more than a decade as a fugitive from war crimes charges.
On February 28th Mr Karadzic gave up, saying he would not come to New York to defend himself.
Bosnian Serb leader Radovan Karadzic has said they were captured on landing but were later kidnapped from a hospital.
Among them are Radovan Karadzic, the former president of the Bosnian Serbs, and Ratko Mladic, once their military chief.
But if the western powers want to stop the Karadzic clan they will have to intervene still more forcibly.
"That I am not guilty, " quipped Mr Karadzic, unable to disguise a smile.
But the television signal does not reach Pale, where Mr Karadzic remains popular.
If the court decides Mr Karadzic has defaulted, the damages could run into the millions of dollars, even a billion.
On March 1st Radovan Karadzic, the former Bosnian Serb leader, opened his defence at his war-crimes trial in The Hague.
ECONOMIST: More arrests and court cases revive bad Balkan memories
International officials have tried to force Karadzic -- wanted for war crimes -- out of power over the past few months.
U.S. officials now hope that Serbian President Slobodan Milosevic can convince Bosnian Serb President Radovan Karadzic to sign a similar letter.
Not so Radovan Karadzic, the former president indicted for war crimes, and Momcilo Krajisnik, the Serb member of Bosnia's collective presidency.
Since the Karadzic arrest in 2008, only two Serbs have been left on the wanted list of The Hague war-crimes tribunal.
ECONOMIST: More arrests and court cases revive bad Balkan memories
If Bosnia is to be held together, the West will have to vanquish not only Mr Karadzic, but also his friends.
Under the peace plan, Karadzic and Mladic are required to step down.
"Serbia has been under tremendous pressure from the international community and EU leaders to deliver Karadzic and Mladic, " said CNN's Alessio Vinci.
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