We construct approximate solutions to the time--dependent Schr\"odinger
equation $i \hbar (\partial \psi)/(\partial t) = - (\hbar^2)/2 \Delta \psi + V
\psi$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and
growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up
to errors whose norms are bounded by $C \exp{-\gamma/\hbar}$, for some $C$ and
$\gamma>0$. Under more restrictive hypotheses, we prove that for sufficiently
small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded
by $C' \exp{-\gamma'/\hbar^{\sigma}}$, for some $C', \gamma'>0$, and
$\sigma>0$.