More than 10,000 people from eastern Democratic Republic of Congo have crossed into Uganda to flee fighting, Ugandan military officials say.
Nkunda says the minority Tutsi community has been excluded
Fighting broke out early on Tuesday between the Congolese army and forces loyal to dissident general Laurent Nkunda, they say.
One Congolese border post is now in the hands of the rebels.
Meanwhile, 11 of Gen Nkunda's men, including two senior officers, have turned themselves over to UN forces.
Some 17,000 United Nations peacekeepers operate in DR Congo, overseeing the peace process after the end of a bloody five-year war in 2002.
Joseph Kabila, who won October's presidential run-off elections, is due to be sworn in on Wednesday as president - the country's first democratically elected leader in 40 years.
Mr Nkunda left the army and launched his own low-level rebellion after the war ended, saying that the country's transition to democracy was flawed and had excluded the minority Tutsi community.
Ugandan army spokesman Lt Tabaro Kiconco told the BBC that the fighting had centred on two border posts about 15km (nine miles) inside Congolese territory, and the post at Runyonyi was currently in the hands of the rebels.
People had begun crossing the border on foot early on Tuesday morning, he said.
The latest violence has flared up in the wake of the presidential elections.
Last week, Gen Nkunda took control of the small eastern town of Sake, 25km (15 miles) west of the provincial capital of Goma, but was eventually repelled by UN peacekeepers.
According to the Congolese army, Gen Nkunda's men surrendered - with their arms - after an appeal from President Kabila, who visited the area recently.
The BBC's Arnaud Zajman in Kinshasa says it is a significant move, as the officers were close to the renegade general.
The head of the UN mission in DR Congo, William Swing, told the BBC he believed the rebels were beginning to understand that they had no choice but to lay down their arms and join the regular army.
The Congolese army is working with UN troops in eastern DR Congo
But our correspondent says Gen Nkunda's continuing rebellion will depend on whether he can still rely on allies outside the country.
In the past, Rwanda has allegedly supported him with men and ammunition.
So far, fighting has been confined to DR Congo itself, but Lt Kiconco warned that the Ugandan army would take serious military action if any fighters crossed the border.
Many of those civilians who have crossed into Uganda have made their way to Kisoro town.
Regional district commissioner David Masereka told the BBC that no facilities had yet been provided for them.
The UN refugee agency and the Ugandan government are sending representatives to assess the situation.