Tuesday's resolution, sponsored by the conservative party led by Nicolas Sarkozy, the French president, had been widely expected to win approval in the National Assembly with opposition Socialists backing it despite concerns about the wording of an eventual law.
France's neighbour, Belgium, is also planning a similar veil ban.
Sharp criticism has accompanied France's nearly year-long debate on banning the face veils, with those opposed saying, among other things, that the entire process has stigmatised the nation's estimated 5 million Muslims - the largest Muslim population in western Europe.
They also say it is a political ploy because only an estimated 1,900 women wear veils that hide the face.
Michele Alliot-Marie, the justice minister who is writing the draft law, said it is the veil that "stigmatises Islam" and "harms the liberty of women on whom it is imposed".
France banned Muslim headscarves and other "ostentatious" religious symbols in classrooms in 2004 after a bitter debate.
|Muslim headscarves and 'ostentatious' religious symbols are banned in French schools [EPA]
The road from a resolution to a bill then a law fully banning face-covering veils is likely to be rocky, too.
Sarkozy's government has been warned by the Council of State, the nation's highest administrative body, that such legislation may not pass constitutional muster, in France or in European forums.
Earlier on Tuesday, a Council of Europe commission opposed a blanket ban on face coverings, saying such a ban would rob women of their freedom of expression and could violate their religious freedoms.
The Strasbourg-based commission also urged Switzerland to end its ban on the construction of mosque minarets as soon as possible.
The Council of Europe, a 47-nation human rights institution, is to discuss the veil issue next month. Its rulings are binding on all Council of Europe member states.