Demolition work began this week on the site off Station Road.
Lidl plans to start building on Monday, January 23.
West Cumberland Farmers (WCF), which will be moving into new premises on the site, will continue trading throughout the process.
The supermarket will create up to 40 jobs. Lidl revealed its plans for a 19,127sq ft store in 2015 and held a public drop-in event.
Some questioned the need for another supermarket in the town, as well as raising concerns about parking.
WCF, the former Fairfield Garage and Travis Perkins store will all be demolished to make way for the supermarket and parking.
A WCF spokesman said: “The project will be completed in two phases, with the new WCF store being built on the site of the old retail filling station while we continue to trade from our existing store.
"On our relocation into the new unit, the existing store will be demolished to make way for Lidl.”
Traffic access to Lidl will be from Station Road, with a fourth exit added to the existing mini roundabout.
There will be parking for 90 cars.
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