Purely non-atomic weak L^p spaces

Let $\msp$ be a purely non-atomic measure space, and let $1 < p < \infty$. If $\weakLp\msp$ is isomorphic, as a Banach space, to $\weakLp\mspp$ for some purely atomic measure space $\mspp$, then there is a measurable partition $\Omega = \Omega_1\cup\Omega_2$ such that $(\Omega_1,\Sigma\cap\Omega_1,\mu_{|\Sigma\cap\Omega_1})$ is countably generated and $\sigma$-finite, and that $\mu(\sigma) = 0$ or $\infty$ for every measurable $\sigma \subseteq \Omega_2$. In particular, $\weakLp\msp$ is isomorphic to $\ell^{p,\infty}$.