  We obtain an asymptotic formula for the first moment of quadratic Dirichlet
$L$--functions over function fields at the central point $s=\tfrac{1}{2}$.
Specifically, we compute the expected value of $L(\tfrac{1}{2},\chi)$ for an
ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as
$g\rightarrow\infty$. Our approach relies on the use of the analogue of the
approximate functional equation for such $L$--functions. The results presented
here are the function field analogues of those obtained previously by Jutila in
the number-field setting and are consistent with recent general conjectures for
the moments of $L$--functions motivated by Random Matrix Theory.
