A day after the jubilation of South Sudan's independence proclamation, the U.S. ambassador to the U.N. warned Sunday of a "real risk" that the north-south peace process could unravel unless outstanding issues such as oil and border demarcations are quickly resolved.

Celebrations rang out Saturday in the South Sudan capital of Juba, the first day of independence after decades of civil war between Sudan's north and south. Some 2 million people died in the most recent war, from 1983-2005.

On Sunday, the capital appeared hungover from its massive celebration, though small groups of people still sang and danced on street corners. The new country's national anthem played from speakers and cell phones.

The joy of independence day temporarily overshadowed the ongoing hostilities between the northern army and southern-allied forces in the northern state of South Kordofan and other violence along the north-south border. The south and north have yet to agree on a demarcated border, and the issue of oil remains contentious. The south has most of the oil but it must move it through the north's pipes.

Dozens of world leaders joined a crowd tens of thousands strong in Juba on Saturday. The American delegation was led by Susan Rice, who told The Associated Press in a phone interview on Sunday that the U.S. government remains "focused on the urgency of resolving" the outstanding north-south issues.

"We feel that until they are resolved, there's a real risk of the process to date beginning to unravel," she said.

Rice said that the U.S. government would remain "very actively involved" in supporting negotiations between Khartoum and Juba. U.S. officials say they hope the talks will restart in the next week through a process led by an African Union panel.

"As wonderful a day as yesterday (Saturday) was ... we are mindful that even as those presidents pledge a commitment to peaceful and cooperative relations, that these issues are such that in the absence of resolution there is a risk of things beginning to disintegrate," she added.

Sudan President Omar al-Bashir, who is wanted for war crimes for his role in the conflict inSudan's western region of Darfur, attended Saturday's ceremony and appealed for the two nations to work to "overcome the bitterness of the past."

Rice dismissed the possibility of U.S. military intervention in South Kordofan, where northern aerial bombardments have driven tens of thousands of black Africans from the Nuba ethnic group into caves for protection from the raids. She noted that the U.S. has not been involved militarily in Sudan and she doesn't foresee that changing in the near future.

She said that given the great urgency of the ongoing crisis, the U.S. is working hard to continue to "engage diplomatically to try to broker (a) cessation" of hostilities.

Southern Sudan voted in a January referendum to break away from the north as part of a north-south peace deal that ended the decades-long civil war in 2005. But the future of the 4,000-square-mile (10,500-square-kilometer) Abyei region, which lies near the north-south border, is still in flux.

South Sudan is expected to become the 193rd country recognized by the United Nations next week and the 54th U.N. member state in Africa.