Conformal Mapping to Unit Disc

Problem: Find a conformal map from the set A = { z = x + iy in C : x^2 + y^2 > 1, x > 0} to the unit disc {z in C : |z| < 1}.

In class we've been through some examples of this form where we construct a conformal map as a composition of mobius maps and other analytic functions. (exp(z), z^n, etc). In each example we used such compositions to map the given sets into the upper half plane { Im(z) > 0} or the half plane {Re(z) > 0}, and then it is easy to map these conformally into the unit disc. So for this set A I imagine I need to do the same thing, but I can't find the right map(s). Help please!