The company said its 90-strong workforce was also in the process of being relocated to sister sites in Portsmouth, Gillingham and Waterford in Ireland.
Unfortunately, the company has been unable to salvage anything from the plant, which will have to be rebuilt following the fire on 26 July. The development is expected to take at least a year.
Divisional managing director Malcolm Ward said: "We have been astounded at the huge level of support we have received from our customers and our Thorpe team and now just one week on, we have been able to re-start production for our customers."
The Nampak division produces packaging for the pharmaceutical industry, with a range including folding cartons, patient information leaflets, self-adhesive labels and printed foils used in blister packs.
At the time, the blaze involved 230 firefighters and was attended to by 19 engines from Surrey Fire & Rescue Services, as well as crews from London and Berkshire.
The company said at the time the fire caused a "major disruption" to production at the carton plant.
Nobody was injured during the incident and the cause of the fire is not yet known.
Nampak's MY Healthcare site relocates work after fire devastates the Surrey plant
Almost 100% of MY Healthcare's work in progress has been relocated following a huge blaze that broke out at its Thorpe site in Surrey at the end of last month.
