  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 $\Hsup s {u(t)} \leq C {{|t|}^{(s-1)+}}$, where
$H^s$ is the standard spatial Sobolev norm. The proof is an adaptation of
earlier work on the nonlinear Schr\"odinger equation which reduces matters to
bilinear estimates.
