An apparatus for machining an article (L) in the form of a slab or the like comprises-in addition to a rotary tool (24) associated with a spindle (22)- a nozzle (26) ejecting water at very high pressure. Both the rotary tool and the nozzle are supported by a carriage (20) sliding along a beam (18) in turn sliding along two shoulders (14) in a perpendicular direction. In the working area (12) and above a tank (30) normally filled with water, the article (L) is supported in a horizontal position not only by an interchangeable grid (32; 52) , but preferably also by a disposable support (34) which prevent the rotary tool (24) from coming into contact with the grid. The disposable support is periodically replaced.