Let M be a real hypersurface with almost contact metric structure $(phi,g,xi,eta)$ in a complex space form $M_n(c)$, $c eq0$. In this paper we prove that if $R_{xi}L_{xi}g=0$ holds on M, then M is a Hopf hypersurface in $M_n(c)$, where $R_{xi}$ and $L_{xi}$ denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field $xi$ respectively. We characterize such Hopf hypersurfaces of $M_n(c)$.