In a statement, the company said 43 staff, who have found alternative jobs, have been allowed to go with a full benefits package. The remaining 55 staff will stop work this week.
South African-owned Nampak has said the restructuring was part of an investment programme that it hoped would enable it to become the lowest-cost packaging producer in the UK.
Over the last few months, it has announced significant capital expenditure on equipment and automation at its purpose-built facilities.
It said that by focusing on its most modern factories that offer the most efficient production, Nampak Cartons would stay ahead of the changes within the industry and satisfy customer requests.
Earlier this year, Nampak announced it was considering closing its Crewkerne factory in Somerset and reorganising administrative roles at its Gillingham site in Kent.
An initial estimate of 17 redundancies at the company's Gillingham site will be "significantly reduced", Nampak said.
The company operates in the plastics and folding cartons in Europe, through its MY Cartons, MY Healthcare and Nampak Plastics divisions.
Nampak Cartons makes 98 staff redundant and closes Crewkerne site
Nampak Cartons has made 98 staff redundant as it announced it will permanently close its Crewkerne, Somerset, carton factory by 20 December.