Works will see the main classroom buildings undergo extensive mechanical and electrical renovation, with new ceilings, floors and decoration, as well as upgraded ventilation, air conditioning and heat recovery systems.
As well as improving building services, the project will create 150 new student places.
The project is part of a wider plan by Bedford Borough Council to switch from a three-tier education system (with lower, middle and upper schools) to a two-tier system (with primary and secondary schools).
Construction will take place over seven phases, with the majority of the school remaining open while work is ongoing. Completion is set for spring 2021. While phases have been planned to minimise disruption as far as possible, hot lunches are off the menu for pupils this autumn term.
The contract was awarded through the Department for Education framework.
Morgan Sindall is also nearing completion on a new £6.8m primary school, Abbey View Primary Academy, at the former Daws Hill RAF site in High Wycombe, which is being delivered for Buckinghamshire County Council.