The last weeks of the war have been marked by a growing chorus of international concern over the fate of Tamil civilians caught up in the intense fighting.
Sri Lanka's military spokesman, Brig Udaya Nanayakkara, says 25,000 civilians have made it out of the conflict zone in the past 72 hours, 11,800 of them on Saturday alone.
However, more remain inside the small area where the Tamil Tigers may be preparing for a fight to the death.
The UN and Western governments have called on Sri Lanka to exercise restraint in its pursuit of a military victory over the Tigers.
British Prime Minister Gordon Brown has become the latest leader to speak out on the issue, declaring on Saturday that there would be "consequences" if Sri Lanka did not work to ensure an orderly end to the conflict.
The final decisive battle 'is reaching its conclusion'
Speaking in Jordan, Mr Rajapakse said: "My government, with the total commitment of our armed forces, has in an unprecedented humanitarian operation finally defeated the LTTE militarily.
"I will be going back to a country that has been totally freed from the barbaric acts of the LTTE," he added.
We have restricted the LTTE to [a] one sq km-like area, so we will mop up and seize the rest of the LTTE cadres and the leadership
Gotabhaya Rajapakse Sri Lanka defence secretary
But the BBC's Charles Haviland, in the capital, Colombo, says a senior government spokesman in Jordan with the president could not confirm that fighting had ended in the north-east.
There are now concerns that the Tamil Tiger leadership is preparing for a last stand, our correspondent says.
The Tigers' leadership has said repeatedly that they will not surrender, and are thought to be keeping thousands of civilians as human shields.
There have also been reports that the Tigers are preparing a mass suicide in the face of a military defeat.
A UN spokesman, Gordon Weiss, told Associated Press the organisation still had "grave fears" for the safety of an estimated 30,000 to 80,000 people still inside the combat zone.
Earlier, Sri Lanka's defence secretary told the BBC that the army was closing in on the remaining fighters for the Liberation Tigers of Tamil Eelam (the full name for the Tamil Tigers).
"We have restricted the LTTE to one square kilometre-like area, so we will mop up and seize the rest of the LTTE cadres and the leadership," Gotabhaya Rajapakse said.
He said the army did not know exactly where to find rebel leader Vellupillai Prabhakaran, but expressed confidence he would be tracked down.
"If he has not committed suicide then he should be there," Mr Rajapakse said.
Prabhakaran began the fight for a separate state for Sri Lanka's minority Tamils in the early 1970s, progressing into a violent civil war in 1983.
The capture of the Tigers' last remaining stretch of coastline earlier on Saturday was hailed as a decisive breakthrough by Sri Lanka's army.
Few images exist of Tamil Tiger leader Velupillai Prabhakaran
For the first time ever the rebels do not have sea access, and the army is "progressing" to clear remaining rebel-held land, Brig Udaya Nanayakkara said.
Speaking after the president's statement, he said military action in the north-east was ongoing, also branding it a "rescue, humanitarian operation".
The Tigers once boasted a deadly and much-feared naval strike capability, often using boats to launch lethal attacks during the long years of their guerrilla campaign.
State TV showed images of explosions and plumes of smoke rising from close to the coastline, as well as images of celebrating troops in coastal areas.
The army said the scenes proved that the Tigers were blowing up their own caches of ammunition, but there is no independent confirmation of that claim.
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