The legislation, which came into effect on Wednesday, imposes a fine of 1,000 kroner (£119) on anyone wearing garments that hide the face.
The 28-year-old woman was fined after fighting with a woman who was trying to remove her traditional Muslim face veil, known as a niqab.
Police were called to a shopping centre in Horsholm, in the northeastern region of Nordsjaelland, where they reviewed the CCTV, according to duty officer David Borchersen.
“During the fight her niqab came off, but by the time we arrived she had put it back on again,” he told the Ritzau news agency.
The woman was informed she would receive a fine in the post, and was told to remove her veil or leave the public space.
“She chose the latter,” Mr Borchersen said.
Both women were charged with breaching the peace.
The new ban has prompted protests and reignited a debate about whether the country discriminates against Muslims.
Although the law does not specifically mention Islamic dress, simply saying ”anyone who wears a garment that hides the face in public will be punished with a fine”, campaigners say the law deliberately targets Muslims.
On Wednesday evening, women wearing niqabs and burqas, which also cover the entire face, gathered to protest in central Copenhagen, where they were joined by dozens of supporters wearing makeshift face coverings. A similar demonstration was held in the city of Aarhus at the same time.
The police did not fine the protesters, since under the new law, those who cover their faces while peacefully protesting are exempt from the ban.
Those who repeatedly flout the ban can be fined up to 10,000 kroner (£930) under the legislation, which was passed in May.
Human Rights Watch has called the rule ”discriminatory” and said it was the “latest in a harmful trend”, after similar bans came into force in other European countries including France, Austria and Bulgaria.
The European Court of Human Rights last year upheld a ban on face-coverings in Belgium, saying the state had the right to impose restrictions that aimed to ensure the principles of “living together”.
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