The invention relates to an adjustable attachment for metal-working lathes and the like, and the error adjusting method made possible thereby for eliminating runout and axial errors in a manner that makes one adjustment independent of the other. The proposed attachment device basically comprises: a first portion which attaches to the headstock spindle of the lathe; a second portion which holds the chuck; and a third portion disposed intermediate the first and second portions, with all three portions being braced together but free to be moved relative to one another in a sideways or radial direction during error adjustment. The relative motion between the first and third portions corrects for axial error and is about a circular path whose center of curvature is on the axis of rotation of the spindle and far enough out to clear the jaws of the chuck; whereas, the relative motion between the second and third portions corrects for runout error and is perpendicular to the axis of the second portion and substantially perpendicular to the spindle axis. Means for clamping the three portions together, once the runout and axial errors have been corrected, is also provided.