  Let $\msp$ be a measure space and let $1 < p < \infty$. The {\em weak
$L^p$}\/ space $\wlp$ consists of all measurable functions $f$ such that
  \[ \|f\| = \sup_{t>0}t^{\frac{1}{p}}f^*(t) < \infty,\]
  where $f^*$ is the decreasing rearrangement of $|f|$. It is a Banach space
under a norm which is equivalent to the expression above. In this paper, we
pursue the problem of classifying weak $L^p$ spaces isomorphically when $\msp$
is purely atomic. It is also shown that if $\msp$ is a countably generated
$\sigma$-finite measure space, then $\wlp$ (if infinite dimensional) must be
isomorphic to either $\ell^\infty$ or $\seq$.
  The results of this article were presented at the conference in Columbia,
Missouri in May, 1994.
