Abstract: This is a system for the air conditioning of rooms in building, which rooms are defined by concrete floor structures with hollow ducts connected in series in parallel with each other and in groups, in order to bring about effective heat exchange between concrete and supply air flowing through each duct group before being fed to the room via a supply air device. The supply air to each duct group is taken via a pipe connection from a main duct for supply air and is evacuated from the room in another way. In order to control the heat absorbtion (heat inertia) of the duct group according to the actual demand for each room so that the air-flows in each of the two connections of the duct group are balanced corresponding to the actual cold/heat demand. Each or some certain duct groups in the room includes a branching device (16) which is located between the main duct (5), or a branch thereof, and a second connecting place (11) to the duct group.