The World Health Organisation said on its website that a second New Zealand case had been confirmed.
The Director-General of Health Dr Ashley Bloomfield said the citizen is a woman in her 30s who had recently returned from a trip to Italy.
The positive test result was formally reported to the ministry at about 6pm yesterday. At that point, health authorities began notifying the woman and her family and started the process of contact tracing.
The woman returned on an Air New Zealand flight NZ0283, from Singapore to Auckland on 25 February.
Bloomfield said anyone who was on that flight and is concerned or wants information should contact the Healthline number (0800 358 5453).
The woman also took two domestic flights - one on 2 March from Auckland to Palmerston North (NZ5013), and on the same day from Palmerston North to Auckland (NZ8114).
Bloomfield said the woman is self-isolating at home and is being monitored. She does not require hospitalisation.
Her partner has also displayed symptoms and is being tested.
Bloomfield said with continued vigilance, the chance of widespread community outbreak remains low.
Contact is being made with the two medical centres where the woman sought advice and treatment to determine whether there is any risk to staff or other people who may have been at the centre at that time. Contact tracing and action will be taken as required.
Two schools have also been notified about the positive test as there is a family member at each of those schools. These schools are Westlake Boys and Westlake Girls' High Schools.
The family members who attend these schools are not symptomatic and are now at home in isolation. They did not travel to Italy, are both well and are being monitored.
New Zealand's first case was a woman who had returned from Iran via Bali. She is recovering in hospital in Auckland.
The ministry said 160 tests have been carried out. All the testing of New Zealanders for Covid-19 were being done in New Zealand.
Air New Zealand says it is working closely with government agencies to identify and contact other passengers on board the international and domestic flights.
The airline's chief medical officer Dr Ben Johnston said the three aircraft will undergo a deep clean using a disinfectant that kills viruses.
Total infections worldwide have topped 90,000, with deaths now over 3100. Nearly all cases and deaths have been in China but the numbers in Iran, Italy and South Korea have continued to grow.