The crew member arrived in New Zealand on December 9 on a flight from the United States.
They tested positive as part of mandatory testing within 48 hours of returning from the country.
The ministry said Air New Zealand had given its assurance the person had complied with all requirements for crew flying to and from the US - and the risk to the general public was considered low.
While overseas, air crew are legally required to wear protective equipment while in direct contact with passengers or if outside of the aircraft or their accommodation.
They must travel directly to their accommodation and cannot leave except for emergencies.
Air crew who fly on "high-risk" routes - which include Los Angeles and San Francisco - also have to follow legal requirements when they return to New Zealand including a mandatory medical examination and Covid-19 test within 48 hours of arrival.
They do not have to go into managed isolation, but have to self-isolate until they produce a negative test.
Covid-19 Response Minister Chris Hipkins said this afternoon the positive case showed the system was working.
"We're seeing evidence of protective measures that have bought a working out with this latest Air New Zealand crew member.
"We know that there is an extra risk for people coming and going who are part of a flight crew," he said.
"We can't have them in two weeks of isolation, or we'd never have any planes coming and going. It just wouldn't be practical.
"This person has been isolating since they arrived back, they got the test as they were expected to [and] they're not showing any symptoms at this point.
"All of the fellow crew members have tested negative, so there's not really any risk of it spreading in the community."
All other crew members have returned negative tests and are in isolation.
The ministry also confirmed three new cases of Covid-19 in New Zealand today, all in managed isolation.
There are no new community cases.
Two of the new cases arrived from the United States, and the other one from the United Kingdom. They have all been transferred to Jet Park quarantine facility in Auckland.
One previously reported case has now recovered, so the total number of active cases in New Zealand is now 56.
There were 4245 tests yesterday.
Previous cases
Yesterday it was announced the crew member arrived in New Zealand on December 9 on a flight from the US.
They tested positive as part of mandatory testing within 48 hours of returning from the United States.
The ministry said Air New Zealand had assured it that the person had complied with all requirements for crew flying to and from the US - and the risk to the general public was considered low.
While genome sequencing has not been finalised, early results showed no link between the crew member's test and any existing New Zealand cases. Comparison with international samples suggested that it originated in the United States, the ministry said.
All other crew members had returned negative tests and were in isolation.
The total number of confirmed cases in New Zealand was 1740.
There were six new cases of Covid-19 in managed isolation announced on Friday.
At that time, there were no community cases and the total number of New Zealand's cases sat at 57.
Of the six new cases, five are active and one was historical:
• One case arrived in New Zealand on November 26, from South Africa via the United Arab Emirates and Malaysia. This person was tested because they were in a travel bubble with two confirmed cases and is at the Auckland quarantine facility.
• One case arrived in New Zealand on December 1, from Sweden via Qatar and Australia. This person was tested because they were in a travel bubble with two confirmed cases, and is at the Auckland quarantine facility.
• One case arrived in New Zealand on December 3, from the United Kingdom via the United Arab Emirates. This person was tested because they were in a travel bubble with a confirmed case, and is at the Auckland quarantine facility.
• One case arrived on December 3 from the United Kingdom via the United Arab Emirates. This person tested positive at routine testing at around day 3, and has been transferred to the Auckland quarantine facility.
• One case arrived on December 3 from the United Arab Emirates. This person was tested because they were in a travel bubble with a confirmed case, and is at the Auckland quarantine facility.
• One case arrived in New Zealand on December 7, from the United Kingdom via Singapore, and was tested upon arrival because they were symptomatic. Serology testing revealed this case is historical and therefore not infectious. This person is at a Christchurch managed isolation and quarantine facility.
From this week, the frequency of updates has been decreased to four times a week.
New Zealand had its first confirmed case of Covid-19 on February 28, and from March 3, daily updates have been provided by the Ministry of Health.
Last Monday, the ministry announced it was cutting the frequency of the updates to four times a week.
The updates are now scheduled for Monday, Wednesday, Friday and Sunday.
There have been no new cases reported in the community since November 21, when the last case linked to the Auckland Defence Force cluster was reported. That person has now tested negative and there are now no active cases listed in the community.
