Assembly (10) for fastening a railway rail (1), comprising a lower platen (11) provided with through holes (111) for anchoring the lower platen to ground (40) by means of anchoring means (15), an upper platen (12) superposable on the lower platen for supporting the rail (1), and a pair of rail fastening clips (142) for fastening the rail to the upper platen (12). The lower and upper platens comprise a pair of corresponding first holes (112, 121) distinct from the through holes (111), for removably securing the upper platen (12) to the lower platen (11) by first fastening means (16) independent of the ground anchoring means (15). The upper platen (12) and the rail fastening clips (142) comprise a pair of corresponding second holes (122, 144) distinct from the first holes and from the through holes, for securing the rail fastening clips (142) to the upper platen (12) by means of second independent fastening means (17). The first holes (121) of the upper platen (12) have oblong shape with a longer axis oriented transverse to the rail (1) so as to allow for lateral adjustment of the upper platen (12) relative to the lower platen (11).