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He is a senior official in the south's ruling party, which is set to lead South Sudan to independence in July.
The two sides fought for decades before agreeing to share power and hold a referendum on southern independence.
Analysts fear the the dispute could reignite the north-south conflict, in which some 1.5 million were killed.
Northern troops seized the territory at the weekend after southern forces had ambushed a convoy of its forces in the area, killing 22 people.
Some 20,000 people have now fled the town, which has been left deserted, aid workers say.
UN human rights chief Navi Pillay said she had received reports that northern forces had been shelling and bombing civilian areas.
On Monday the UN said the town had been set ablaze, while gunmen were looting property following the northern takeover.
Mr Biong Deng is originally from Abyei, which both sides claim.
Its status was left undecided in the 2005 peace deal and a referendum, due in January, on whether the area should be part of the north or south has been postponed indefinitely.
Aid 'jeopardised' "We had hoped that we could form two viable states in good relationship with each other, but those in Khartoum do not seem interested in peace," Mr Biong Deng said in his resignation statement.
"But with war crimes being committed in Abyei at the hands of the [President Bashir's] National Congress Party, I could not in good faith continue to take part in such a government."
His party, the Sudan People's Liberation Movement, has urged the northern troops to withdraw from the town.
The US envoy to Sudan, Princeton Lyman, has warned that the takeover could jeopardise possible debt relief worth billions of dollars.
He also said that Washington would find it difficult to remove Sudan from its list of state sponsors of terrorism unless troops were withdrawn from Abyei.
Under these sanctions Sudanese companies are banned from using US currency - a major obstacle to international trade.
The US has previously suggested that a peaceful transition to independence for the south and a negotiated solution to the separate conflict in Darfur could normalise relations.
Sudan: A country divided
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The great divide across Sudan is visible even from space, as this Nasa satellite image shows. The northern states are a blanket of desert, broken only by the fertile Nile corridor. Southern Sudan is covered by green swathes of grassland, swamps and tropical forest.
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