  We prove a quenched central limit theorem for random walks in i.i.d. weakly
elliptic random environments in the ballistic regime. Such theorems have been
proved recently by Rassoul-Agha and Sepp\"al\"ainen in [10] and Berger and
Zeitouni in [2] under the assumption of large finite moments for the
regeneration time. In this paper, with the extra $(T)_{\gamma}$ condition of
Sznitman we reduce the moment condition to ${\Bbb E}(\tau^2(\ln
\tau)^{1+m})<+\infty$ for $m>1+1/\gamma$, which allows the inclusion of new
non-uniformly elliptic examples such as Dirichlet random environments.
