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A project which was due to be completed last year is expected to finish this summer.
Work to widen a busy 2.5 mile stretch of the A21 between Tonbridge and Pembury has already been delayed due to asbestos removal.
More than 35,000 tonnes of contaminated earth had to be removed.
It was initially scheduled to end in December last year. Now it has been announced that Pembury Walks and Half Moon Lane will shut from Wednesday to allow builders from Balfour Beatty to complete the scheme.
The roads will be closed at their junctions with A21 Tonbridge Road, until the summer.
The alternative route for Half Moon Lane is via Alders Road, A228 Maidstone Road/Pembury Northern By-Pass, A264 Pembury Road and A21 Pembury By-Pass. For Pembury Walks drivers will be diverted via Old Church Road, A228 Pembury Northern By-Pass, A264 Pembury Road and A21 Pembury By-Pass.
The dual carriageway aims to help reduce congestion while a new pedestrian and cycle route is being built to provide a safe route between Tonbridge Station and Pembury Hospital.