Ban on Veils for Women

According to the New York Times,  two women were fined for wearing the niqab, shown below, an Islamic facial covering that leaves only the eyes uncovered.  This is newly against the law in France.

"The fines were the latest examples of the country’s struggle to assimilate its Muslim population of more than five million. They came less than a week after a ban on street prayers aimed at Muslim worshipers spilling into the streets of Paris and other cities entered into force across France."

No one has been arrested or fined for praying in public.

Would such a law be constitutional in our country?  Can we ban articles of clothing or praying in public?