The bill, which has received overwhelming support in opinion polls, must now be ratified by the senate in September to become law.
The opposition Socialist party, who originally wanted the ban limited only to public buildings, boycotted Tuesday's vote.
France's highest administrative body warned in March that the bill could be found to be unconstitutional and therefore thrown out.
Estelle Youssouffa, Al Jazeera's correspondent in Paris, said: "The Council of the State warned the government that the French law and the EU law could find this bill unconstitutional, as it violates human rights and religious freedom."
The bill makes it illegal to cover the face anywhere in public and those caught wearing a full veil would face fines of $190 or be ordered to enrol in a "citizenship course".
Men who force their wives or daughters to wear the full veil face a fine of up to $37,754 and a one-year jail term, according to the draft legislation.
Nicolas Sarkozy, the French president, said earlier this year that the full veil "hurts the dignity of women and is not acceptable in French society".
Life in France is "carried out with a bare face", Michele Alliot-Marie, the justice minister, said last week as she opened the debate in the National Assembly.
Face-covering veils "call into question the idea of integration, which is founded on the acceptance of the values of our society", Alliot-Marie said.
The main body representing French Muslims says face-covering veils are not required by Islam and not suitable in France, but it worries that the law will stigmatise Muslims in general.
The veil is widely seen in France as a sign of extremism and an attack on women's rights and secularism, a central value of modern-day France.
Critics say the ban is a ploy to attract far-right voters.
Our correspondent said the Muslim community in France is uncomfortable with the bill and feels it is stigmatising the whole community.
"French Muslims say less than only 2,000 women use the veil and are seen as ultra-orthodox minority and do not represent the whole community," she said.
In April, politicians in Belgium's lower house voted almost unanimously to ban the wearing of face veils in public places.