The document reasserts Serbian claims of sovereignty over the ethnic-Albanian majority province of Kosovo.
Kosovar ethnic Albanians, who seek independence, did not take part in the two-day referendum.
Final official results were not immediately available, but the Center for Free Elections and Democracy, which monitored the referendum, estimated that more than 51 percent of Serbia's 6.6 million registered voters backed the constitution.
Voter turnout was estimated at more than the required 50 percent.
Prime Minister Vojislav Kostunica said the constitution marked a new start for Serbia after Montenegro declared independence and left its union with Serbia earlier this year.
"Serbia is stronger with the constitution, with a constitution that clearly says that it is a democratic state ruled by law, a constitution which, by invoking international law, reaffirms that Kosovo is part of Serbia," Kostunica said.
President Boris Tadic said Serbs had now left behind the old constitution that dated back to former President Slobodan Milosevic's regime. Tadic noted that the new constitution sets the stage for early general elections expected before the end of the year.
(compiled from agency reports)
Spotlight On Kosovo
THE WORLD'S NEWEST NATION? The region of Kosovo has a population of more than 2 million, some 90 percent of whom are ethnic Albanians. It was one of the poorest regions in the former Yugoslavia, but has considerable mineral wealth and an enterprising population, many of whom work abroad but keep close contact with Kosovo. All ethnic Albanian political parties seek independence on the principles of self-determination and majority rule. They feel that Serbia lost its historically based claim to what was its autonomous province under the 1974 constitution by revoking that autonomy in the late 1980s and then conducting a crackdown in 1999 that forced some 850,000 people to flee their homes.
Since NATO's intervention that year to stop the expulsions, Kosovo has been under a UN administration (UNMIK). The UN has begun to gradually transfer functions to elected Kosovar institutions. The primary Serbian concerns are physical safety for the local Serbian minority, a secure return for the tens of thousands of Serbian displaced persons, and protection for historic Serbian religious buildings. The main problems affecting all Kosovars, however, are economic. Until Kosovo's final status is clarified and new legislation passed and enforced, it will not be able to attract the investment it needs to provide jobs for its population, which is one of the youngest and fastest growing in Europe. Prosperity is widely seen as the key to political stability and interethnic coexistence in Kosovo, as is the case in much of Southeastern Europe.
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