The radiation-selective absorber coating (20) has two barrier layers (24a, 24b), an IR-reflecting layer (21) arranged thereon, an absorption layer (22) arranged above the IR-reflecting (21) and an antireflection layer (23) over the absorption layer (22). The absorber tube (13) is a steel tube (1) with the radiation-selective absorber coating (20) applied to the outside thereof. In the process of coating the absorber tube (13) a first oxide barrier layer (24a) is applied to a steel tube by thermal oxidation; a second barrier layer (24b) is then applied by physical gas phase deposition of silicon with supply of oxygen; the IR-reflecting layer (21) is then applied by gas phase deposition of gold, silver, platinum or copper; the absorption layer (22) is then applied by deposition of aluminum and molybdenum; and a final antireflection layer (23) is applied by deposition of silicon with supply of oxygen.