  Let $\bar\gamma$ be a link in a Seifert fibered space $M$ over a hyperbolic
$2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of
closed geodesics $\gamma$ in $\mathcal O.$ We prove that the complement
$M_{\bar\gamma}$ of $\bar\gamma$ in $M$ admits a hyperbolic structure of finite
volume and give combinatorial bounds of its volume.
