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Spatial regularity properties of certain global-in-time solutions of the Zakharov system are established. In particular, the evolving solution $u(t)$ is shown to satisfy an estimate $|u(t)|_{H^s} leq C |t|^{(s-1)+}$, where $H^s$ is the standard spatial Sobolev norm. The proof is an adaptation of earlier work on the nonlinear Schrodinger equation which reduces matters to bilinear estimates.