Assume your surface is conformally equivalent to a disc. 
Let $u$ be conformal factor; it has to go to $\infty$ as you go to absolute.
Since Gauss curvature is nonnegative, the function $u$ has to be kind-of-subharmonic
--- if I remember right Gauss curvature is something like $\Delta\tfrac1u$.
The later contradicts maximum principle.