Sudanese President Omar al-Bashir has threatened to close "forever" an oil pipeline that carries oil from South Sudan to Sudan's Red Sea coast.
He said on Monday that Sudan will stop the flow of oil if South Sudan supports rebels operating on Sudanese soil, speaking on state TV.
The Sudanese army is fighting a rebel insurgency in at least three regions.
Despite the South's independence in 2011, tensions over oil and land disputes have continued.
"I now give our brothers in South Sudan a last, last warning that we will shut down the oil pipeline forever if they give any support to the traitors in Darfur, South Kordofan and Blue Nile," President Bashir said on state television, referring to rebels operating in these regions.
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An umbrella rebel group called the Sudan Revolutionary Front (SRF) has launched attacks on several towns, briefly occupying the major city of Um Rawaba in central Sudan in April.
The group, which hopes to topple President Bashir's government, withdrew from Um Rawaba, but held onto the town of Abu Kershola, in the neighbouring oil-rich region of South Kordofan.
SPLM-North rebels joined the Darfuri rebel groups, Jem and two main factions of the Sudan Liberation Army (SLA), to form the Sudan Revolutionary Front last year.
President Bashir delivered his speech live on Sudanese state TV following the army's announcement that it had recaptured Abu Kershola from the rebels.
"Thank God, Abu Kershola has been liberated," Defence Minister Abdelrahim Mohamed Hussein said, quoted on state TV.
Meanwhile, a spokesman for the rebels told AFP news agency that the fighters withdrew from Abu Kershola to ease a government blockade on its residents.
Relations between the two countries have been fraught since 2011, when the South became independent after decades of civil war.
Key issues related to oil production, territorial disputes and border demarcation remain unresolved.
South Sudan took with it nearly three-quarters of Sudan's oil production when it declared independence. The two sides fell out over how much the South should pay to export its oil through Sudanese pipelines.
At the height of the dispute last year, the South shut down its entire oil output, badly hitting both struggling economies.
Oil started flowing again last month after both sides struck a deal in the Ethiopian capital, Addis Ababa, in March, helping to ease tensions.
They also agreed to withdraw troops from their border area.
However, the latest violence has put further strain on relations.
President Bashir warned on Monday that "failure to abide by any agreement will nullify the nine accords" agreed in March.