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In an email to parents today rector Richard Hall said the pupil, who showed up for class this week, was now isolating and had returned a negative Covid-19 test this morning.
The Otago Daily Times understands police are investigating the breach.
Mr Hall said they had sought advice from Ministry of Health, the Ministry of Education, as well as police.
The Ministry of Health had advised the school he was considered a "low risk" as he was not presenting any symptoms and was understood to have not visited any places of interest in the Auckland or Waikato region.
Mr Hall said the pupil travelled from his home in Auckland, where he had been since the lockdown began, to Dunedin on Friday.
He stayed with family for the weekend and spent Sunday night at the school's hostel.
He then arrived at school for first period yesterday.
Yesterday at 9am the school was advised the pupil may not have had a travel exemption.
"We immediately implemented our isolation policy for the student, and organised a Covid 19 test, the results of which, we have had back this morning and they are negative.
"The student is isolating away from the school and hostel. He will not return to either the school or the hostel until we have confirmation from the Ministry of Health that it is safe for him to do so."
The school had received advice from the Ministry of Health that it and the hostel could run as it had been under Level 2 protocols.
Mr Hall said maintaining learning momentum was important "this close to the end of term".
"We understand that this news may be of concern to you.
"Please be assured that the health of our school community is very important to us and we will be following all protocols.
"We do ask you to respect the privacy of the student, and to continue to maintain the supportive environment that we are proud of at OBHS."
Mr Hall said if parents or their children at the school required support they could contact the school.
The ODT has sought comment from the school.