The device for carrying out the present method is configured as a machining unit having a device (30) measuring the outside of a workpiece in a contactless manner. Said measuring device is arranged on the machine such that it opposes the surface of the workpiece to be measured at least some times. According to the present method, the workpiece is machined while clamped into the machining unit, and the machined surface of the same is measured in a contactless manner with the measuring device, specifically while the workpiece is still clamped into the machine. This method is suited for machining workpieces, particularly for grinding the surface of non-round workpieces.