The cost of repairing the ancient bridge and surrounding pathways at Tarr Steps will run into thousands of pounds.
Somerset County Council and Exmoor National Park say the landmark on Exmoor has been affected by landslips, fallen trees and erosion from recent flooding.
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The whole region staying clear overnight with warm temperatures.
Temperatures remain high up to 12C inland.
Until now the ground has just been too waterlogged to let them out to graze.