  In this paper it is shown that for $\Omega\in L\log L(\mathbb{S}^{d-1})$, the
rough maximal singular integral operator $T_\Omega^*$ is of weak type
$L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $\Omega\in
L^\infty(\mathbb{S}^{d-1})$, it is shown that $T_\Omega^*$ is of weak type
$L\log\log L(w)$ with weight dependence
$[w]_{A_1}[w]_{A_{\infty}}\log([w]_{A_{\infty}}+1),$ which is same as the best
known constant for the singular integral $T_\Omega$.
