A method for filling up a storage tank (e.g., a vehicle tank) with a gaseous, pressurized medium, in particular in the form of hydrogen, in which a supply tank system for storing the hydrogen is connected with the storage tank to be filled by way of a tank feed line and a fueling valve, wherein, prior to filling up the storage tank with aforesaid medium with the fueling valve closed, a flow of the medium for cooling the tank feed line is guided through the tank feed line at a predefinable target temperature, and removed from the tank feed line through a line that branches away upstream from the fueling valve. The disclosure further provides for a fueling facility for filling up a storage tank.