  We show that, given a non-degenerate, finitely connected domain $D$, its
boundary, and the number of its boundary components, it is possible to compute
a conformal mapping of $D$ onto a circular domain \emph{without} prior
knowledge of the circular domain. We do so by computing a suitable bound on the
error in the Koebe construction (but, again, without knowing the circular
domain in advance). As a scientifically sound model of computation with
continuous data, we use Type-Two Effectivity.
