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The comments were made by the US ambassador to the UN, Susan Rice, following a Security Council meeting on the issue.
South Sudan has accepted the border roadmap proposed by the African Union, but Sudan is refusing to do so.
The two countries came close to an all-out war this year over the disputed border and oil revenues.
South Sudan won independence from Sudan last year, ending decades of fighting between the mainly Muslim north and Christian and animist south.New deadline
Speaking in New York, Ms Rice said Khartoum's refusal to sign to the roadmap plan "risks the resumption of outright conflict".
She stressed that it also "calls into question Khartoum's seriousness" to resolve the issue.
Despite South Sudan's acceptance of the plan, Ms Rice said that Washington was "deeply concerned by the apparent lack of urgency" shown by both sides.
The original 2 August deadline set by the UN failed to produce an agreement.
However, both sides are being kept under pressure to clinch a deal by a new deadline - 22 September.
Khartoum and Juba last month held three weeks of talks in the Ethiopian capital Addis Ababa.
They resulted in a provisional oil payments deal, which is subject to further talks on security.
The talks resumed earlier this week. A summit between the presidents should follow to make the deal official.
BBC.
BBC.