The province earlier this month resolved its crisis over the elections and is forming a government team led by President-elect Sergei Bagapsh. There have been comments from both Georgia and leaders of the separatist republic in support of negotiations.
Council President Cesar Mayoral, Argentina's UN ambassador, read a statement stressing the need to revive the more than decade-old political process: "[Council members] underlined the need for renewed efforts to achieve a lasting political solution to the conflict, based on the relevant Security Council resolutions, and expressed their hope for an early resumption of the dialogue between the two sides in order, inter alia, to prevent instability and to build mutual confidence."
The Security Council supports a settlement based on Abkhazia remaining a part of Georgia, which the separatists refuse. Abkhazia broke off high-level contacts with Georgia in the summer to protest a maritime incident but returned in December to weekly meetings involving Georgia and the United Nations on smaller issues.
UN envoy Tagliavini told reporters after briefing the council that new contacts are important. But after the turmoil in Abkhazia, she said, no one should expect dramatic results.
"There has been more than three months of internal political crisis [in Abkhazia]," Tagliavini said. "This has strained the patience of the people. It has brought a lot of tension to the area, and I think the people need to digest what they have just gone through."
Security Council diplomats told RFE/RL that they are placing hopes on the next round of high-level talks to be held by the Group of Friends of the Secretary-General for Georgia.
Envoys from the Group -- from Russia, the United States, Germany, France, and Britain -- are due to meet in early spring in Geneva to discuss ways of advancing the Georgia-Abkhaz process. Yesterday's Security Council statement urged both sides to participate.
Meanwhile, Abkhaz President-elect Bagapsh reaffirmed the republic's claim to independence in Moscow yesterday.
But the council statement later in the day reaffirmed its commitment to the sovereignty and territorial integrity of Georgia "within its internationally recognized borders."
Bagapsh added in Moscow that progress on restoring economic ties could improve the dialogue between Sukhum and Tbilisi.
Tagliavini said that if both sides can make progress on security guarantees, that can serve as a way of advancing political dialogue.
The UN envoy told RFE/RL that Bagapsh's experience in the drawn-out political process with Georgia could help move talks forward.
"He has participated in the peace process," Tagliavini said. "He knows the issues. [He's] not an unknown figure for us, so I think the language will be found for discussion, and he will certainly be engaged himself in the peace negotiations -- directly or indirectly."
The Security Council on 28 January is expected to extend the UN peace mission in Georgia by six months. The mission consists of 130 uniformed personnel, including 119 military observers and 11 civilian police. It was established in 1994 after an accord reached in Moscow ended the fighting.