Valve seats are machined at the end of an axially deep, narrow cavity of a workpiece having an axially clampable surface within the cavity adjacent one or more valve seats. The method comprehends: (a) supporting a rotary driven spindle assembly of sufficient length to extend through the axial extent of the cavity and reach the valve seat, the spindle carrying a cutting tool at its extremity, the support being in a nonrotatable housing effective to fit closely to the interior of the cavity (within a tolerance of 0.01 inches) and effective to move along the axis of the cavity to bring the cutting tool into machining contact with the valve seats; (b) securing a radially rigid but axially resilient diaphragm to the end of the housing, the diaphragm extending transversely across the interior of the cavity; (c) after inserting the housing and spindle into the the cavity so that the cutting tool is proximate the valve seat but out of contact therewith, axially clamping a portion of the diaphragm remote from the spindle, against the axially clampable surface; and (d) moving the rotary driven spindle assembly axially toward the valve seat to conduct machining contact of the cutting tool therewith, the axially resilient diaphragm providing accurate supporting alignment for the tool under even high speed machining conditions.