In a machine for covering a toroidal body with layers of elastomeric material wherein a cutting and stitching apparatus is provided for severing a scrap portion of a sheet of material (32) from a portion of the sheet that has been applied to a toroidal core (30) and stitching the applied portion to the core (30). The apparatus (16,18,20,22) includes actuators (45,54) for positioning the cutters (104) and the stitchers (105) in operative position on the outside periphery and the inside periphery of the core (30) with the rotatable cutters (104) in proper position for effecting a cut through the material without scoring the core (30) or other underlying material. The cutting and stitching apparatus (16,18,20,22) is rotated about the core (30) as the rotating cutter (104) severs the material. The cutter (104) is programmed to retract as the stitchers (105) continue to stitch the edge of the applied material to the core (30).