Rail vehicle having a device for finishing the running surface of track rails

A rail vehicle, comprising:

• a chassis frame;
• a set of wheels coupled to the chassis frame;
• and
• a milling device coupled to the chassis frame for finishing a running surface of track rails during travel of the rail vehicle along the track rails, wherein the milling device includes a milling head and a framework arranged on the chassis frame with a milling head receptacle that is displaceable vertically and transversely in relation thereto, the milling head receptacle being supported for vertical guiding via a guide shoe on the running surface and being supported for transverse guiding in parallel to a milling head shaft via a guide stop laterally on a railhead, wherein the milling head receptacle forms a transverse carriage which supports the guide shoe and the guide stop, the transverse carriage having a guide path that is parallel to the milling head shaft and that is vertically displaceable and provided on a side of the milling head receptacle facing toward the guide shoe and wherein a diameter of the milling head is less than or equal to half of a wheel diameter of the wheel sets.