The Schools Development Authority gave its unanimous approval on Tuesday. The new building will replace a structure that has been in a state of decay for years.
The construction plans call for tearing down most of the high school building to construct new wings, while keeping its grand main entrance, clock tower and auditorium.
The school serves more than 1,800 students.
The SDA initially proposed repairing the structure, which would have cost about $127 million. But officials say the construction project will fully remediate the health and safety concerns of students and staff.