An apparatus for producing seamless plastic tubes comprises an injection head to which molten plastic material is fed by an extrusion device. The injection head has a plate-shaped distribution device 13a, 13b having an intake opening 20 into which the feed passage 17a, 17b for the molten plastic material opens. In the distribution device, distribution passages 14 extend in a star configuration from the intake opening 20 and extend perpendicularly to the axis of the injection head, wherein each distribution passage 14 has a straight radial passage portion 14r and three branch passages 14z which branch therefrom and which each go into arcuate passage portions 14k which, in the region of the periphery of the distribution device, communicate with the annular gap 25 of the injection head, with the adjacent ends at a respectively identical angular spacing, and they each introduce a respectively identical volume flow into the annular gap 25.