In a fossil-fuelled steam generator (1) with a flue, the containig wall (2) of which takes the form of gastightly interconnected pipes (4), which are substantially vertically arranged and through which the medium can flow from bottom to top, the invention proposes that the pipes (4) have a greater inside diameter (d1) in a first or lower part (5) of the flue than the pipes (4) in a superimposed second part (7) of the flue. This ensures that the pipes (4) are reliably cooled. In addition, a greater than average heating of individual pipes (4) does not lead to unacceptable temperature differences between the outlets of the pipes (4).