There is disclosed an assembly (100) for supporting and fastening to the ground (10) rails railway and/or subway lines with cushioning of vibrations, comprising a plurality of slabs (1) comprising fastening means (11) in the form of a parallelepiped formed by a rigid central structure (15) fixed to the plane (10) bearing said slab and by a cushioning frame (12) which surrounds said central part (15), said means being housed in a through cavity (8) formed by two recesses (4) of two adjacent slabs (1) formed on the sides (5) of said slabs (1) perpendicular to the longitudinal axis (3) of said rails (21).