The invention relates to a method for filling up a storage tank (e.g., a vehicle tank) (20) with a gaseous, pressurized medium, in particular in the form of hydrogen, in which a supply tank system (2) for storing the hydrogen is connected with the storage tank (20) to be filled by way of a tank feed line (3) and a fueling valve (4), wherein, prior to filling up the storage tank (20) with aforesaid medium with the fueling valve (4) closed, a flow of the medium for cooling the tank feed line (3) is guided through the tank feed line (3) at a predefinable target temperature, and removed from the tank feed line (3) through a line (5) that branches away upstream from the fueling valve (4). In addition, the invention relates to a fueling facility (1) for filling up a storage tank (20).