The assembly (1) has a main container (2) including a dispenser endpiece (10) with a distal dispensing orifice (11) and a lateral dispensing orifice. An auxiliary container (3) includes a stopper (6) with a filling opening through which the endpiece is introduced for filling the auxiliary container. An obturator is placed in the filling opening, so as to disassemble the stopper and fill the auxiliary container through the lateral orifice. A ring (14) is moved from a closed position to an open position during insertion of the endpiece, where the stopper is closed during removal of the endpiece. An independent claim is also included for a method for filling the auxiliary container.
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