An automatic machine and an automatic method for grinding the edges of glass sheets (1). The machine is provided with a machine body (2b) with motorized support and conveyance rollers or belts (3b), an input conveyor (2a) with motorized support and conveyance rollers or belts (3a), an output conveyor (2c) with motorized support and conveyance rollers and belts (3c); there are also at least two means for conveying the glass sheets (1), a lower one (100) and an upper one (200), which actuate respectively the synchronous motions about a lower axis (X1) and an upper axis (X2), which engage and convey the glass sheets (1), which are interfaced alternately, for example the odd sheets with the lower conveyance means (100) and the even sheets with the upper conveyance means (200).
