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, such as the niqab or the burqa, "hurts the dignity of women and is not acceptable in French society".
'A bare face'
In crafting the bill, officials have sought to avoid singling out Muslims.
While the proposed legislation is colloquially referred to as the "anti-burqa law," it is officially called "the bill to forbid concealing one's face in public".
The text refers neither to Islam nor to veils - leading to an often surreal disconnect between the text and discussion in parliament about it.
While officials insist the law against face-covering would apply to everyone, not just Muslims, they cite a host of exceptions, including masks for health reasons, for fencing, for carnivals and festivals.
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.
As legal reasoning, she invoked the notion of public policy doctrine, the country's moral and social rules.
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.
Socialist Jean Glavany, one of the few politicians to offer strong criticism of a ban, said dwelling on questions of French identity and whether burqas are welcome in France "is nothing more than the fear of those who are different, who come from abroad, who aren't like us, who don't share our values".
Failing to push through the ban would also be a humiliation for the conservative government of Nicolas Sarkozy, the French president, which has devoted much attention to a bill that would affect only an estimated 1,900 women.