Bell Bridge at Sebergham collapsed into the River Caldew six weeks after being hit by the December 2015 storm.
Work on the new bridge and neighbouring road is due to start on 17 July, Cumbria County Council said.
The project was due to start in June but was delayed due to the need to divert overhead power cables.
Keith Little, cabinet member for highways and transport, said: "The loss of Bell Bridge had a significant impact on the local community and I'd like to thank them for their patience and understanding during this period."
Bell Bridge was built in 1772 and was named after a family from a nearby farm.
The council said it was looking at 300 bridges this year that require some form of repair after Storm Desmond.
More than 130 were repaired last year.
Work to replace a 245-year-old bridge which was destroyed by Storm Desmond should be completed by November, a council has said.