The route is closed in both directions between Thistly Cross roundabout and Abbotsview Junction with winds of 84mph expected.
A diversion route is in place via the A199. This is expected to add approximately 10 minutes to affected journeys.
The Met Office has issued a Red weather warning for wind.
Drivers of high sided vehicles, motorbikes, caravans, and cars with trailers or roof boxes should divert via the A68.
Tommy Deans, BEAR Scotland’s Network Manager for South East Scotland, said: "“This section of the A1 includes an exposed bridge over the River Tyne, so with gusts up to 84mph forecast this evening we’ve closed it to all traffic for safety, in line with the bridge’s wind management plan.”
“We’ve now closed the A1 to high sided vehicles for safety with winds gusting over 50mph.
“All road users are advised to take care and check for restrictions before travel. If we experience the very high wind speeds that are forecast later today, this will result in a full closure of the A1.
“Please check Traffic Scotland for the latest travel advice before setting off and consider whether your journey is essential.”