A milking system comprises a milking device for milking a milking from a dairy animal during a milking session, and is provided with a control unit for the milking system, milking cups, a first milk jar for receiving the milking, a first pumping device for pumping the milking from the first milk jar to a first milk pipe, a second milk jar in flow communication with the first milk pipe for receiving the milking from the first milk jar therefrom, and a second pumping device for pumping the milking from the second milk jar into a second milk pipe. The first pumping device is configured to pump the milking at a first flow rate and the second pumping device is configured to pump the milking at a second flow rate, which is lower than the first flow rate. The milking system furthermore comprises a heat- exchanging system in flow communication with the second milk pipe for receiving the milking pumped in the second milk pipe and for bringing the received milking to a desired temperature. The capacity of the milking system is maintained by pumping the milking directly after a milking, but by collecting this milking in the second milk jar and pumping it relatively slowly and steadily, the milk quality is maintained more reliably. In addition, a downstream heat exchanger can then be made smaller. Also, it is then easier to change the desired temperature.
