In a Facebook Live update, Ardern said it needed to be determined whether the person picked up the virus overseas and had a late-onset, which she said isn't common but is possible, or they picked up the virus during managed isolation.
That information should be known that within the next 24 hours, she said.
She also said there is no need to move up alert levels following Sunday's developments.
"Remember those times when we've had to move up those alert levels, that's been where we haven't been able to identify what's happened. We know what's happened here. We're just narrowing down exactly how it happened."
Ardern also urged Kiwis to be kind to one another, after noticing comments online where people appeared to "blame" those with the virus.
"This happens almost every time we have a Covid case. Sometimes there's a tendency to blame and the message I really want to share here is the most important thing for us is that people when they are unwell they continue to get tested."
She said people are less likely to get tested if they feel like they'll be attacked if the result is positive.
"We really need people not to pile in on anyone if they do the right thing by having a Covid test," she said.
"We need people to keep feeling like they can come forward. It's just so important that we keep that kindness, that we just treat people the way we would like to be treated if we were in that circumstance, imagine if your family member tested positive."
She also addressed a misconception among some people that if they have the Bluetooth function of their phone turned on and have the Covid app, they don't need to scan the Covid QR code.
"Unfortunately, that's not the case. We need you to scan everywhere you go. The Bluetooth element just tells us if you've been in close contact with someone else who might in the future test positive. That enables us to identify people who've been in close proximity. It doesn't tell us where you've been," she said.
"The fantastic thing is we have someone here who has. So we are reaching out to all of those venues that we need to and all of the people who've scanned into those venues."
She also discussed calls to close New Zealand's borders entirely.
"We can't do that. As a New Zealand citizen, you have the right to come back to New Zealand. If we denied you the right to come back to New Zealand, we would be making you stateless," she said.
"If we were to close our border entirely, that means we wouldn't get in our imports, we wouldn't be getting in goods into the country, we wouldn't get medical supplies."
The Ministry of Health also reported on Sunday that there had been eight new cases Covid-19 in managed isolation since Friday.
