A film laminate wound up to form a film roll, from which bags with an integrated handle are produced in a VFFS form-fill and seal bag machine. The film laminate is produced by coating a film web of an outer film with a lamination adhesive and applying a material web of a sealable film to the adhesive layer. The material web is pressed together with the film web and adhesive layer in a lamination gap between a lamination roller and a pressing roller. A reinforcement strip passed to the lamination gap contacts the adhesive layer, between the film web and the sealable film. The circumference surface of the pressing roller has a depression that is coordinated with the film thickness and the film width of the reinforcement strip, so that a uniform pressing pressure prevails in the lamination gap, over the entire width of the laminate.