  We show how to express a conformal map of a general two connected domain in
the plane such that neither boundary component is a point to a representative
domain which has the virtue of having an explicit algebraic Bergman kernel
function. We shall explain why the representative domain is the best analogue
of the unit disc in the two connected setting. The conformal map will be given
as a simple and explicit algebraic function of an Ahlfors map of the domain
associated to a specially chosen point. It will follow that the conformal map
can be found by solving the same extremal problem that determines a Riemann map
in the simply connected case.
