The Open Treasure exhibition provides a new route to the monks' dormitory and the great kitchen - one of the last surviving monastic kitchens in England.
Money has been spent installing a lift near the monks' staircase leading to the sleeping quarters, which were constructed between 1398 and 1404.
A new cathedral shop has also been built and the restaurant refurbished.
The project included new lighting and underground heating elsewhere in the cathedral.
The exhibition will be unveiled on 23 July.
A cathedral spokesman said: "[The] new world-class exhibition route will provide access to previously hidden spaces within the cathedral's magnificent claustral buildings."
The cathedral is regarded as one of the finest examples of Norman architecture and is a UNESCO World Heritage Site.
A £10m revamp of Durham Cathedral is to provide public access to previously hidden spaces.