City council members in the southern Zealand municipality were deadlocked 12-12 on the proposal, which was set forth by Jonas Kristinsson, a member of the national ruling party Venstre.
The tie vote meant that the proposal was scrapped, much to the relief of Faxe's mayor, Knud Erik Hansen.
“It [the ban] is unnecessary. It sends the wrong signal and I'm not even sure we could even [legally] do it,” he told broadcaster TV2.
Despite being against the ban, Hansen said he's not a fan of the face-covering garments worn by some Muslim women.
“Children in daycare institutions and schools, and people who receive in-home help shouldn't encounter people who cover themselves up. But on the other hand, there are jobs within the municipality that don't entail contact with the problem, and that's where [the ban] could be a problem,” he said.
Kristinsson, the architect of the proposal, said there are no current reported problems involving employees wearing Muslim garb in Faxe, located some 70 kilometres southwest of Copenhagen. But it could be an issue in the future, he said.
“I don't know how our society will look in five or ten years. Therefore I think we should listen to the concerns of residents. I hear from many residents who fear running into a municipal employee dressed like that,” he told broadcaster DR.
Danish politicians have targeted face-covering veils before. In 2009, the Conservative party suggested a nationwide ban on burqas. Only the anti-immigration Danish People's Party (DF) supported the idea, which was quickly shot down after the justice ministry advised that it would violate the constitutional right to the freedom of religion.
Following the European Court of Human Rights' decision to uphold France's ban on wearing face-face veils in public, DF spokeswoman Pia Kjærsgaard said “it is completely obvious” that Denmark should do the same.
The party, however, presented no formal proposal to parliament.