An apparatus for making a cold-pilger-rolled tube has two grooved pilger rolls rotatable about respective axes to compress a workpiece moving in a travel direction against a mandrel extending in the direction between a nip formed between the rolls and anchored upstream of the nip in a thrust block. Workpiece thickness is adjusted on the fly by measuring a wall thickness of the workpiece downstream in the direction from the roll as the workpiece passes through the nip and for displacing the thrust block and mandrel in to the travel direction as the workpiece passes through the nip in accordance with the measured wall thickness.