  Let $p:\R\to(1,\infty)$ be a globally log-H\"older continuous variable
exponent and $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular
integral operator $S$ is bounded on the weighted variable Lebesgue space
$L^{p(\cdot)}(\R,w)=\{f:fw\in L^{p(\cdot)}(\R)\}$ if and only if the weight $w$
satisfies \[ \sup_{-\infty<a<b<\infty}
\frac{1}{b-a}\|w\chi_{(a,b)}\|_{p(\cdot)}\|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty
\quad (1/p(x)+1/p'(x)=1). \]
