Centering machine for workpieces, namely optical lenses with curved optical surfaces, with two centering spindles (22, 30), the centering spindle shafts (24, 34) of which, each driven in rotation and accommodated in centering spindle housings (26, 32), are axially aligned with respect to a centering axis and each accommodate a cup-shaped clamping bell (36, 38) at mutually facing ends, a stroke device, by means of which one of the centering spindle shafts (24, 34) can be adjusted axially in a sensitive manner along the centering axis with respect to the other of the centering spindle shafts (24, 34) in order to align one of the workpieces between the clamping bells (36, 38) in a bell clamping process, in which the clamping bells (36, 38) are applied to the workpiece with slight pressure in a probing phase before the workpiece is automatically displaced in a direction transverse to the centering axis in an alignment phase as a result of the curvature of its optical surfaces until it has assumed a position between the clamping bells (36, 38) which permits the minimum distance between the clamping bells (36, 38) under the given geometric conditions, a clamping device for applying a clamping force to one of the centering spindle shafts (24, 34) in order to clamp the workpiece aligned between the clamping bells (36, 38) in a clamping phase following the alignment phase, and at least one processing unit (58, 60) movable relative to the centering axis with a tool for edge processing of the workpiece clamped between the clamping bells (36, 38), wherein both the stroke device and the clamping device are arranged coaxially with respect to the centering axis and wherein the clamping device has an annular piston which is attached to the axially adjustable centering spindle shaft and forms an annular effective surface on its side facing away from the clamping bell end of the axially adjustable centering spindle shaft, which annular effective surface defines an annular chamber in the centering spindle housing, via which the annular piston can be acted upon pneumatically in order to generate the clamping force. two centering spindles (22, 30), the centering spindle shafts (24, 34) of which, each driven in rotation and accommodated in centering spindle housings (26, 32), are axially aligned with respect to a centering axis and each accommodate a cup-shaped clamping bell (36, 38) at mutually facing ends, a stroke device, by means of which one of the centering spindle shafts (24, 34) can be adjusted axially in a sensitive manner along the centering axis with respect to the other of the centering spindle shafts (24, 34) in order to align one of the workpieces between the clamping bells (36, 38) in a bell clamping process, in which the clamping bells (36, 38) are applied to the workpiece with slight pressure in a probing phase before the workpiece is automatically displaced in a direction transverse to the centering axis in an alignment phase as a result of the curvature of its optical surfaces until it has assumed a position between the clamping bells (36, 38) which permits the minimum distance between the clamping bells (36, 38) under the given geometric conditions, a clamping device for applying a clamping force to one of the centering spindle shafts (24, 34) in order to clamp the workpiece aligned between the clamping bells (36, 38) in a clamping phase following the alignment phase, and at least one processing unit (58, 60) movable relative to the centering axis with a tool for edge processing of the workpiece clamped between the clamping bells (36, 38), wherein both the stroke device and the clamping device are arranged coaxially with respect to the centering axis and wherein the clamping device has an annular piston which is attached to the axially adjustable centering spindle shaft and forms an annular effective surface on its side facing away from the clamping bell end of the axially adjustable centering spindle shaft, which annular effective surface defines an annular chamber in the centering spindle housing, via which the annular piston can be acted upon pneumatically in order to generate the clamping force.