The legislation in front of the lower house will make it illegal to wear the niqab or burka anywhere in public.
A vote on the proposed legislation will be taken next week before a full Senate vote in September.
The veil ban, which has the backing of President Nicolas Sarkozy, is also winning support in other parts of Europe.
Opponents of the ban point out that only a small number of women wear the full veil. Others say existing laws that prohibit wearing a disguise in public should be applied. Meanwhile many people born and raised in Western Europe see the full veil as an assault as Western culture and values.