The closure will be in place between Woodside Cafe at the Bonsall turn, to Grangemill at the junction with the B5056.
Signed diversions will be in place.
A council spokesperson said: “Our tree inspectors will assess the situation once the storm has passed over the area and, if safe to do so, the road will re-open at 16:00 on Friday.
“We realise that this will be a real inconvenience to those who regularly use this road, but public safety comes first.”