The boundary dispute is important because under a peace deal agreed between the north and south in 2005, Abyei will hold a referendum in 2011 on whether to retain special status within north Sudan, or join the south, where a simultaneous vote will be held on independence.
Malik Agar of the former rebel Sudan People's Liberation Movement (SPLM), which heads an autonomous regional government in the south, said that violence was to be expected but that neither side would allow it to get out of hand.
"The decision, of course, will not be without disappointment to either party," he said.
"We expect some violence may be there, but the two parties are prepared to quell any violence."
Ghazi Salahaldin, of the northern-based National Congress Party of Omar al-Bashir, the president, said: "We have committed ourselves to the arbitration and we are going to accept the ruling."
Abyei town was razed in the clashes in May 2008 and tens of thousands of people were left homeless, most of them ethnic Ngok Dinka loyal to the south, who have long had tense relations with the district's Messeria Arab nomads, regarded as loyal to the north.
The UN special representative expressed concern last week that armed groups involved in the fighting last May were reported to be in and around Abyei.
But on Monday Ashraf Qazi said that all unauthorised military units were pulling out of the district.
"We believe that action has been taken to ensure that the area will be clear of such elements, and that is very reassuring," he said.
Under the 2005 peace deal, the only forces allowed in Abyei are special joint north-south police and military units.