A spokeswoman said the ministry won't comment on whether the results were back before or after the quarantine ended but said the woman went into self-isolation at a private residence.
No contract tracing was being done because "all people on the flight (including this person) had entered into managed isolation on arrival into New Zealand".
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A list of confirmed cases on the Ministry of Health's website said she had arrived in the country on March 25 but the ministry has since confirmed that date was incorrect.
The Herald has asked for more information on the case, including when she developed symptoms - if any - and if the test was a "weak positive", or thought to have come in the late stages of her illness.
The average incubation period for the virus that causes Covid-19 is thought to be about five days. But there are reports of some people receiving positive tests for the virus long after they are thought to have recovered.
The case follows the announcement this week that a Marist College student had just received a "weak positive" confirmation that she had the virus, despite a cluster breaking out at her school more than a month ago.
It was thought she was in the very late stages of her illness and was no longer infectious.
Today's second case was linked to the St Margaret's Hospital & Rest Home in Auckland.
The person is a household contact of an earlier case linked to St Margaret's. They have been in self-isolation since that case was notified.
No press conference was held this afternoon but the Ministry of Health released the details.
Two people are in hospital and there are 102 active cases across the country. There have been a total 1494 cases - 1144 confirmed and 350 probable.
The past three weeks have single-digit case numbers each day, with two days in a row of zero cases in the past week.
This is the final release of public data before Cabinet's decision tomorrow on when to move down to level 2.
On Wednesday a student at Marist College was reported as testing positive for the virus, despite the Auckland school's Covid-19 cluster breaking out more than a month ago.
Since then all students have supposedly been in self-isolation and lockdown. The student in question had only had one symptom - a loss of smell - and wasn't they had the virus until the school's students went for widespread testing.
Director-general of health Dr Ashley Bloomfield said that test was a "weak positive" so it was almost certainly very late in their illness.
But the significance of the weak positive test so late was not fully understood.
He said it was unlikely the student was still infectious, but would remain in isolation and be tested again next week.