Two of the Dunedin man's family members, including the student, had been in isolation after showing symptoms.
The man had only developed symptoms five days after returning, so there was no risk to others on the plane he travelled on, Director-General of Health Dr Ashley Bloomfield said earlier on Tuesday.
A statement from Logan Park High School said the student did not attend school on Tuesday as a precaution after his father's positive test.
The student was tested yesterday and the school was told on Tuesday afternoon he had tested positive for the respiratory virus.
"While this isn't the outcome we were hoping for, there is a plan in place which has been activated," the school said.
"We believe that the family have done everything according to the instructions of the Ministry of Health. The parent and student are in isolation.
"We are supporting the family and continue to be in regular contact with them."
Logan Park High School had been advised to close for two days and close contacts of the student - staff or students who have been less than two metres away and/or have had more than 15 minutes' contact - need to self-isolate.
"We are keeping our parent, staff and student community informed and realise this is an anxious time for all concerned."
The school was undertaking all steps advised by the Ministry of Health and will update the community when more information becomes available.
"They will be able to return to school as usual when the school reopens," said a statement from the Southern District Health Board.
"Southern DHB public health staff have already started contact tracing to identify individuals who may have come into close contact with the parent and will be working with the school and family to identify any close contacts of the student over the next 24 to 48 hours."
Medical Officer of Health Susan Jack added: "We appreciate that there are many concerns in the community at the moment. It is most important in these challenging times that we care for and support one another, and continue to play our part in preventing the spread for this illness".
The Ministry of Health said the school would be carefully cleaned before reopening.
"The school will not open until cleared to do so by health officials."
The ministry said these steps would limit the chances of further cases.
"The Ministries of Education and Health are working closely with Public Health services in Dunedin and the school to help minimise disruption, progress contact tracing and reduce the possibility of spread."
Dunedin mayor Aaron Hawkins urged locals to remain calm.
"This is a very serious situation as we've seen internationally and as we are now seeing in isolated cases around the country, so it's understandable that people are anxious about it.
"But a number of organisations have been preparing for these sorts of outcomes and at this point we don't have community transmission and the goals is still to delay that as long as possible."
Two confirmed cases of the virus in Dunedin made it a local problem, rather than a national or international issue, Hawkins said.
"The best response is to stay calm, be sensible and be prepared.
"We're all responsible, regardless of how fit and healthy we are. We're all responsible for looking after those who might not be in quite as fortunate a position."
Two more Wellington cases
Two more Wellington coronavirus cases were also confirmed on Tuesday - a man in his 30s and his father, aged in his 70s, who recently returned from the United States, but neither require hospital care.
They arrived in Auckland on American Airlines flight AA83 from Los Angeles on Saturday March 14.
The younger man was unwell on the flight, the father became unwell about a day later. The pair then flew from Auckland to Wellington on Air New Zealand flight NZ419 in seats 1B and 1C.
The two men had been picked up from Wellington Airport, then spent the weekend at home. They called ahead before going to the doctor, which was the correct thing to do, Bloomfield said.
Those full-time workers eligible for the package will receive $585 per week from the Government, paid in a lump sum package of just over $7000 covering a 12-week period.
The Government is also raising benefits by $25 a week, starting April 1, and doubling the winter energy payment.
This is just the first tranche of the Government spending response – the rest will be unveiled during May's "recovery" Budget.
The package comes as part of "the most significant peace-time economic plan in modern New Zealand history".