The ban makes it illegal for Muslim women to wear a burka (a head-to-toe garment) or a niqab (a face covering) in public.
Headscarves, turbans and Jewish skull caps will not be affected by the law.
Interestingly, this new law, popularly dubbed as "Burqa Ban" is seen as targeting Muslim women in the country who choose to wear a particular type of clothing.
Rights groups say such bills discriminate against Muslim women who wear face-concealing veils.
Because the law allows people to cover their faces for a "recognizable goal", such as in cold weather, Justice Minister Soeren Pape Poulsen said, police officers will have to use "common sense", according to the AP.
People will only be allowed to cover their faces for a "recognisable purpose" when the law comes into force.
Wearing a burqa, which covers a person's entire face, or the niqab, which only shows the eyes, in public will lead to a fine of 1,000 kroner ($156, 134 euros).
Fines will range from 1,000 Danish crowns (160 dollars) for a first offence to 10,000 crowns for the fourth violation.
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The justice ministry and the police now will write more detailed guidelines.
Denmark becomes the fifth European country on Thursday to ban Burqa and Niqab.
Human rights campaigners say the law will have particularly negative consequences for Muslim women, limiting their identity and freedom.
Two women wearing the Islamic niqab stand outside the French Embassy during a demonstration in London, on April 11, 2011.
Van Gulik added that the law fails "abjectly" if its goal is to protect women's rights.
The Danish government had proposed to ban the full-face veil earlier this year in February.
"With a ban on covering the face, we are drawing a line in the sand and underlining that in Denmark we show each other trust and respect by meeting face to face", Poulsen said.
Women wearing the Islamic veil sit in the audience seats of the Danish Parliament. "But some people use them to promote an ideology that, if they are successful, will mean many more lose their freedoms", Martin Henriksen, the Danish People's Party spokesperson on immigration issues, said, according to The Local DK.