The move also paves the way for the complete deployment of the European Union's 2,000-strong peace and justice mission to Kosovo, which is to oversee Kosovo's "supervised" independence.
The constitution declares Kosovo a secular republic and "an independent, sovereign, democratic, unique and indivisible state".
The text also expresses Kosovo's determination "to build a future... as a free, democratic and peace-loving country that will be a homeland to all of its citizens".
Addressing parliament, Hashim Thaci, Kosovo's prime minister, said: "The constitution clearly defines our hope, our optimism, our belief and our rights, which are strongly supported and keep us united."
"The constitution is our will and legitimacy. It is a seal of the state of Kosovo. This is another historic step forward in building of our stable state and democratic governance in an independent, sovereign and proud Kosovo."
Despite the vote, many questions remain over how Kosovo's transition will proceed, after Serbia's ally Russia last year blocked the adoption of a UN Security Council resolution endorsing the EU takeover and a UN plan for independence.
The UN mission says it is "awaiting instructions" from headquarters on how to proceed with what UN officials now describe as its "reconfiguration" rather than transition.
|"Kosovo will not be a member of the United Nations... and as such it will not belong in the world community of sovereign nations"|
Vuk Jeremic, Serbia's foreign minister
It is certain to remain in some form, and retain some powers, under Security Council resolution 1244.
Pieter Feith, the EU's international civilian representative in Kosovo, said on Monday that he needed "more visibility" on which powers the UN mission would retain.
Kosovo's parliament declared independence from Serbia on February 17 and has since been recognised by 38 countries, including the United States and most of the EU.
Fatmir Sejdiu, the country's president, appealed on Wednesday to Kosovo's 120,000 remaining Serbs to accept the new constitution, and to become a "bridge" between Kosovo and "our neighbour, Serbia".
Meanwhile, Vuk Jeremic, Serbia's foreign minister, said on Wednesday that Kosovo faced a "free fall to failure" if it stuck to its course of self-declared independence from Belgrade.
"Kosovo will not be a member of the United Nations... and as such it will not belong in the world community of sovereign nations," he said.
Addressing a conference in Athens on Balkan security, Jeremic said: "It will remain unattractive to foreign investment, unresponsive to the rule of law, and incapable of preventing its free fall to failure without the engagement of Belgrade."
Jeremic said Serbia would call for a majority vote in the UN general assembly to seek the legal opinion of the International Court of Justice on the issue.
"I think the vast majority of countries that are member-states of the UN would demonstrate their discomfort at the flagrant violation of the UN charter," he said
Lulzim Basha, Albania's foreign minister, who argued with Jeremic during much of the two-hour session, insisted that Serbia's efforts would be futile.
"In every forum, Vuk, you have tried to [prevent Kosovo's independence] but you failed," he said.