  We prove the existence of nontrivial standing wave solutions of the complex
Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi
|^\alpha \phi $ with periodic boundary conditions. Our result includes all
values of $\theta $ and $\gamma $ for which $\cos \theta \cos \gamma >0$, but
requires that $\alpha >0$ be sufficiently small.
