A B road often used by drivers while the A66 is closed has also been shut to traffic.
The B6276 has been closed from Brough to the county border near Middleton-in-Teesdale.
National Highways said the road was closed between the A685 and the A67 as the snowfall was concentrated in the area.National Highways area team crews are on scene with winter treatment vehicles working to clear and treat the carriageway.
However, it added, forecasts predict that snowfall will continue in the area throughout the morning.
Cumbria police are also on scene assisting to clear the traffic caught within traffic.
Westmorland & Furness Council said the B6276 was closed to avoid motorists attempting to get around the A road closure by cutting through trying to use unsuitable minor roads and getting stuck.
