Conformal Flatness of 2-manifolds

Question

So, due to the existence of isothermal coordinates, all 2-manifolds are conformally flat. The consequences of this are a bit confusing to me- this means one can conformally map, for instance, the sphere to some surface with identically zero curvature everywhere. What does this surface look like? Is it an infinitely large sphere? (Arbitrarily large radius would imply arbitrarily small curvature, yes?)