Wates was originally lined-up for the work on the Blashford Tower but failed to agree a final price with the council.
Wates landed a £90m deal with Camden in January 2019 to reclad five towers across the estate in the wake of the Grenfell disaster.
But Wates and Camden parted company last year after failing to agree final terms.
Due to its different structure a separate two stage design and build contract has been awarded for Blashford Tower so that work can start on the building sooner than if it was part of the overall Chalcots buildings tender.
In the first stage – beginning next month – Ccuncil Officers will work with GRAHAM and technical experts to finalise the design details and test how the works will be carried out.
In the second stage of the contract – from July 2022 to December 2023 – the contractor will source materials for the works, set up the site and carry out the works, including installing an A1 rated cladding system, new windows, curtain wall and the other works needed to deliver the highest standard of safety for residents.
Councillor Meric Apak, Cabinet Member for Better Homes, said: “This is a significant step forward in delivering a new standard of resident safety for the Chalcots and I am pleased that we have reached this important milestone for Blashford with the appointment of GRAHAM.
“I share the disappointment that residents have felt about delays and so I know that they too will be pleased of this appointment and to see works getting underway.”