Prime Minister Mette Frederiksen said the mutated virus posed a "risk to the effectiveness" of a future Covid-19 vaccine.
Coronavirus cases have been detected in mink farms in Denmark's northern Jutland region, and in other parts of Europe, for several months.
But cases are spreading fast in Denmark, and five cases of the new virus strain were found on mink farms. Twelve people had become infected, the authorities said.
Denmark is the world's biggest producer of mink fur.
Minks at more than 1,000 farms are to be culled. Police Chief Thorkild Fogde said it would be a "very large undertaking".
As we reported earlier, in Norway, the new mutation of SARS-CoV-2 was found; it has never been observed in the country before – it is easier to get infected with it than with the usual form.
According to scientists, the mutated virus is called M439K and belongs to the genetic subgroup of the coronavirus B.1.5. The surface of this protein differs from the common SARS-CoV-2. It has shorter spikes that help the virus to attach to the cell and infect it. The new type of coronavirus can easily bind with the cells in the body.