The wave two school project for the Department of Education will see the firm replace 800-pupil Wrotham School to provide 265 more places from September 2027.
The final negotiated price is £8m above initial estimates for the project which now has full planning.
The buildings to be demolished are situated to the northwest and southeast corners of the built-up area of the school campus. These will clear the way for a new three-storey building and sports pitch.
The project will be delivered in four phases. To ensure continuity of the education activity, temporary classrooms will be brought on site and the existing buildings to the northwest will remain until the new facilities are completed.