Results of the poll are expected on the morning of March 5 at the earliest.
Candidates from more than a dozen parties took part in the vote, which is not recognized by Georgian authorities.
Abkhaz President Sergei Bagapsh, speaking after casting his ballot in the capital Sukhumi, said the vote is part of the nation's independence drive.
He said the international community should understand there are proper democratic institutions in Abkhazia, including an opposition, independent media, and fair elections.
Abkhazia and another Georgian region, South Ossetia, have run their own affairs since breaking away from the central government in the early 1990s.
They have not gained international recognition, and Tibilisi has pledged to bring them back under central control.
Georgian Condemnation
Abkhazia's election was condemned today by Georgian President Mikheil Saakashvili. "The number of people who were expelled from Abkhazia ranges between 400,000 to 500,000," he said in televised remarks. "Given these circumstances, Any attempt to legalize the lawlessness that is taking place in Abkhazia today is unacceptable and will never be acceptable to Georgia and the international community."
"We want peace, not revenge," he added. "We do not want violence, we want reconciliation."
Georgian Prime Minister Zurab Noghaideli also dismissed today's elections, saying they were "not only illegal, but immoral."
"Nevertheless, we are offering peace talks to the de-facto leadership of Abkhazia -- without any postponement, delay, or precondition," he said. "[These talks], in the long run, should become foundation of full, and peaceful resolution of this conflict. We are offering wide autonomy to Abkhazia."
(Reuters, ITAR-TASS)
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