Let M be a complete rotational hypersurface of a space form with constant scalar curvature S. In this paper we classify these hypersurfaces in the cases of R and H , determine the admissible values of S in each of the three spaces and give a geometrical description of the hypersurfaces according to the values of S. In the case of S we find examples of embedded hypersurfaces with constant S∈( n−2/ n−1, 1), which are not isometric to product of spheres.