The mailpiece sorting machine includes a bin sorting conveyor (1) that acts under the control of a control unit (4) to move the bins (3) over sorting outlets (S1, S2, Sn) having removable storage trays (5). The control unit is suitable for detecting (20) that a current mailpiece has a destination address that does not correspond to an available sorting outlet in the machine. If such a situation is detected, a manipulator robot (R) comes to replace the tray of a sorting outlet with an empty tray (or with a tray already containing mail for the same destination address) into which the current mailpiece is sorted, while the replaced tray is placed on standby on a rack (8).