The 45-year-old crossing requires vital repair work because the joints on the viaduct are approaching the end of their useable life and need to be replaced.
The layout of the road and nature of the work means it can only be carried out safely if one of the A19 carriageways is closed.
By shutting the road fully in one direction, Highways England's team of engineers can work around the clock to complete the job as quickly as possible.
It will be shut northbound from Friday (6 March) until Monday at 6am.
The other scheduled closures are as follows:
20-23 March (Friday 9pm to Monday 6am) southbound closure
27-30 March (Friday 9pm to Monday 6am) southbound closure
17-20 April (Friday 9pm to Monday 6am) northbound closure
Senior project manager Ben Dobson said: "We understand a weekend closure will cause disruption, and thank people for their patience.
"Safety is our top priority and it's crucial we take measures to stop the structure deteriorating to a point that it could become a safety issue.
"Traffic is likely to be at its busiest from 10am to 6pm on Saturday and Sunday, so we would advise drivers to allow plenty of time for their journeys."
The main northbound diversion is the A66 eastbound to A66/A1032 (Cannon Park), Newport Bridge and rejoining the A19 at A1046 (Haverton Hill/Portrack). The southbound diversion is the reverse of the northbound closure.
The work is weather dependant, so three further contingency weekends have been pencilled in for 15-18 May, 29 May - 1 June and 5-8 June.
For more details about the closures visit https://highwaysengland.co.uk/projects/a19-teesside-weekend-closures/ or follow Highways England's regional Twitter feed at @HighwaysNEAST.