For grinding use on the base underside of continuous in-track welded rail type track defining a railroad right of way, for removing rail excess weld metal from the underside of the rail base, at the weld joint thereof, after ballast has been sufficiently removed from between the ties at the joint involved, in which the apparatus comprises a cart that is wheeled to ride on and extend crosswise of the track, and defines at one end a carriage way that extends to one side of the right of way, with the carriage way being equipped with a carriage which carries a post structure that is adjustable lengthwise of same, and that carries an adjustably mounted grinding head that rotatably mounts on the upper side of same a grinding wheel that is applied, when in its operative relation, to the underside of the rail base, from outside the right of way and under the rail involved, in which the grinder head may be raised by adjusting the post structure lengthwise thereof to bring the uppermost portion of the rim of the grinding wheel against the underside of the rail base joint at the gauge side edge of the rail, with the carriage being movable crosswise of the track to shift the grinder wheel across the rail base underside for grinding purposes, both being repeated as needed to grind off from the rail base underside the excess weld metal.