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The three-day lockdown commenced on Sunday night, and the rest of the country has also been placed under heightened restrictions because of the fear that the new cases could be a more contagious strain of the virus.
This is the first lockdown that New Zealand has had to enforce for six months.
The country's Covid-19 minister Chris Hipkins said that the people who had tested positive are a family of three living in Auckland - New Zealand's largest city - and that they could not yet explain where the cases came from.
He said that the authorities are looking to trace anyone that the family might have come into contact with.
Prime Minister Jacinda Ardern had been due to attend an LGBT pride event in Auckland, but has cut short that trip to return to the capital Wellington for discussions about how her government will deal with the situation.
She held an urgent meeting with other politicians in her cabinet before announcing the lockdown in Auckland.
Ardern said that they were being cautious until the details of the outbreak become clear, and they discover whether the cases are linked to another variant of the virus.
In January, the country recorded its first cases of Covid-19 for two months after a 56-year-old woman returned from Europe carrying the virus.
New Zealand has been widely praised for the response taken to the worldwide pandemic, and has managed to keep cases and deaths down to nearly zero for some time.
Sporting events and concerts have been able to take place, as well as citizens returning to work and school.
They have also received their stock of the Pfizer/BioNTech vaccine ahead of schedule, and are set to start vaccinating the population of around five million on February 20.
Speaking at a press conference earlier today, Ms Ardern said: "I'm asking New Zealanders to continue to be strong and to be kind.
"I know we all feel the same way when this happens. We all get that sense of 'not again',
"But remember, we have been here before and that means we know how to get out of this again, and that is together."
Even though there is a mandatory quarantine period of two weeks, the virus has managed to enter the country on a few occasions before now, and officials are investigating how it happened on this occasion.
Mr Hipkins said: "New Zealand has kept Covid-19 contained better than almost any other country. But as we have kept saying, there is no such thing as no risk."
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