A machine for processing mail comprises a mechanical jogging device between an unstacking head (3) and a conveyor path (1) for conveying a stack (2) of postal items. The jogging device comprises inclined rollers (8) rotated so that a postal item of the stack, on arriving vertically over the rollers, is capable of dropping edge-on, under the action of gravity, onto the rollers and of being driven towards a jogging margin (7) while simultaneously being moved towards the unstacking head by the effect of the rollers rotating so that its free face comes substantially into contact against the unstacking head.