  Let $C_\Gamma$ be the Cauchy integral operator on a Lipschitz curve $\Gamma$.
In this article, the authors show that the commutator $[b,C_\Gamma]$ is bounded
(resp., compact) on the Morrey space $L^{p,\,\lambda}(\mathbb R)$ for any (or
some) $p\in(1, \infty)$ and $\lambda\in(0, 1)$ if and only if $b\in {\rm
BMO}(\mathbb R)$ (resp., ${\rm CMO}(\mathbb R)$). As an application, a
factorization of the classical Hardy space $H^1(\mathbb R)$ in terms of
$C_\Gamma$ and its adjoint operator is obtained.
