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Dr Burrows had already been working as interim principal at the school since December, when former principal Andy Kai Fong took up a new role in Dubai.
Haeata board chairman Sam Uta'i said Dr Burrows had made a positive impression on students, staff and the wider community, and proved to be an "outstanding" candidate.
"During the recruitment process, we worked with staff and students from all age groups to find out what they wanted to see in their new principal," she said.
"It quickly became clear that Peggy had earned the support of our staff, and the trust of our students."
The board initially planned to use term one and two to recruit a new principal but had unanimously agreed to appoint Dr Burrows much sooner than expected.
Dr Burrows was Rangiora High principal from 2003 until 2015, when she became embroiled in an employment issue.
In 2017, she was awarded $150,000 for wrongful dismissal.
Said Ms Uta'i: "Peggy has thrown herself into the role in the very short time she has been at Haeata and has already expanded or created new relationships with many of our key stakeholders.
"We're pleased she is now able to continue on that good work."