We study isometric immersions f : M-n -> Rn+1 into Euclidean space of dimension n + 1 of a complete Riemannian manifold of dimension n on which a compact connected group of intrinsic isometrics acts with principal orbits of codimension one. We give a complete classification if either n >= 3 and M-n is compact or if n >= 5 and the connected components of the flat part of W are bounded. We also provide several sufficient conditions for f to be a hypersurface of revolution.812471479