A composite yet monolithic abrasive material-removing lap includes a sintered porous matrix of ceramic or metal material and a quantity of an impregnating material substantially completely filling the pores of the sintered porous matrix preform at least throughout an effective region of the lap that is to come in contact with a workpiece to be acted upon by the lap. The lap is made by first slip-casting a preform, then freezing and freeze-drying the same, followed by lightly sintering the dry preform into a porous matrix preform, bringing the temperature of the preform to above the temperature at which the impregnating material is flowable, and filling the pores with the impregnating material.