Sudanese aircraft have bombed the town of Bentiu in South Sudan, three days after South Sudan pulled out of a disputed oil field in the region.
Several bombs were dropped near a key bridge and in a market, killing at least one child and injuring several civilians.
Sudan and newly-independent South Sudan have accused each other of launching fresh attacks on their territories. Neither side shows any sign of bowing to global pressure to return to the negotiating table.
South Sudan said Sudanese troops attacked settlements about 10km on its side of the border and carried out air raids in a range of areas including its oil-producing Unity state.
Sudan denied the accusations but said it had repelled a "major" attack by SPLM-N rebels in South Kordofan state, on its own side of the border. Sudan routinely says the rebels are controlled by the South.
Tensions have mounted since South Sudan declared independence from Sudan in July last year, under a peace settlement that ended decades of civil war between the two sides.
In the worst fighting since the split, South Sudan earlier this month seized the disputed oil-producing territory of Heglig, raising fears of a return to all-out war - then announced it had started withdrawing on Friday, following sharp criticism from United Nations Secretary-General Ban Ki-moon.
On Saturday, US President Barack Obama urged the leaders to restart stalled negotiations over a series of territorial and oil-related disputes, saying there was still a chance to avoid war.
But there was no letup in the rhetoric over the weekend.
The South's Paul described the Sudanese attacks as "a serious invasion of our territory".
Sudan's State Oil Minister Ishaq Adam Gamaa told Reuters the chance of the sides reaching a settlement soon was now "very remote" and said Khartoum would probably demand compensation for damage to Heglig before returning to talks.
Sudan lost about 40,000 barrels per day of output because of the fighting, he said, but added the country had enough reserves to last up to six months before the impact would be felt in its refineries.
Any return to widespread fighting would have a devastating impact on both oil-dependent countries and push refugees and fighters into the surrounding region.
The fighting has already shut down most of the oil production that fuels both Sudan and South Sudan's struggling economies.