conformal map from a disk minus a radial segment to unit disk

Question

I need to find a conformal map from the disk minus a radial segment $\Omega$ to the unit disk $\mathbb{D}$, where $\Omega = \{ z \in \mathbb{C}: |z|<1, z \notin [\frac{1}{2},1) \}$.

I have tried looking at rotations, exponential maps and mobius transforms, but non of these give the required mapping.