  Let $\chi$ be a non-real Dirichlet character modulo a prime $q$. In this
paper we prove that the distribution of the short character sum
$S_{\chi,H}(x)=\sum_{x< n\leq x+H} \chi(n)$, as $x$ runs over the positive
integers below $q$, converges to a two-dimensional Gaussian distribution on the
complex plane, provided that $\log H=o(\log q)$ and $H\to\infty$ as
$q\to\infty$. Furthermore, we use a method of Selberg to give an upper bound on
the rate of convergence.
