Jacinda Ardern said on Saturday the new measures, including the closure of public venues, would start at 6am on Sunday.
Health officials, who could not immediately confirm how the person got infected, said genome sequencing of the new infection was underway.
The newly infected patient developed symptoms on Tuesday and is thought to have been potentially infectious since last weekend, officials said.
The person, a 22-year-old man who is the older sibling of a south Auckland high school student, had visited several public venues during that period.
“Based on this, we are in the unfortunate but necessary position to protect Aucklanders again,” Ms Ardern said.
The prime minister added: “We must assume that it is the more infectious variant that we continue to deal with.”
The level 3 stay-at-home restrictions in Auckland mean people can leave home only for essential shopping and essential work, Ms Ardern said.
Measures in the rest of the country would be tightened to level 2, including limits on public gatherings.
The move comes a fortnight after Auckland’s nearly 2 million residents were plunged into a snap three-day lockdown when a family of three were diagnosed with the more transmissible UK variant of Covid-19.
Health authorities were trying to find out whether the new case was linked to the earlier February cluster, now at 12 infections.
New Zealand, one of the most successful developed nations in controlling the spread of Covid-19, has seen just over 2,000 cases of the virus and 26 deaths since the start of the pandemic.
A marquee Twenty20 cricket clash in Auckland between New Zealand and Australia, scheduled for Friday, is now set to be played in Wellington without crowds, New Zealand Cricket said.
The new restrictions also complicated the America’s Cup Event yacht race scheduled to start on 6 March in Auckland’s harbour. America’s Cup Event said on Twitter it was working “through the implications”.