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Assume your surface is conformally equivalent to a disc $D$ and $e^\phi$ be the conformal factor. From completeness, $\phi(x)\to\infty$ as $x\to \partial D$. Gauss curvature can be expressed as $K=-\tfrac12{\cdot} e^{-\phi}{\cdot}\Delta\phi$. Thus, $\Delta\phi\le 0$. The later contradicts maximum principle.