  Let $\ag$ be an affine Lie algebra, and let $\Ua$ be the quantum affine
algebra introduced by Drinfeld and Jimbo. In [Kas94] Kashiwara introduced a
$\Ua$-module $V(\lambda)$, having a global crystal base for an integrable
weight $\lambda$ of level 0. We call it an {\it extremal weight module}.
  In [\S13]{Kas00} Kashiwara gave a conjecture on the structure of extremal
weight modules. We prove his conjecture when $\ag$ is an untwisted affine Lie
algebra of a simple Lie algebra $\g$ of type $ADE$, using a result of
Beck-Chari-Pressley [BCP].
