A delivery head for delivering a milk portion is connected via a connection line to a milk container in a refrigerator. A rinsing agent source is connected via a first line to a connection part in the connection line. A first on-off valve is inserted in the first line. The connection part is also connected via a second line and a second valve to a fresh air source. The container, the valves, and the connection part are installed in a refrigerator. Through this implementation, complete emptying of the connection line and the delivery head of rinse water after the rinsing procedure, as well as efficient rinsing during the rinsing procedure, are achieved.