A cutting apparatus for cutting tubular banding material in a banding machine includes as guide assembly for guiding tubular banding material in a flattened condition along a guide path from an inlet end to a cutting station, and a single cutting blade mounted for movement in a transverse cutting path across the flattened material at the cutting station. A pinch mechanism is provided for pinching opposite sides of the flattened material together immediately adjacent the cutting station to hold the material securely against bunching up while it is being cut.