We investigated the effect of alloying on the upper critical field $H_{c2}$ in 12 $MgB_2$ films, in which disorder was introduced by growth, carbon doping or He-ion irradiation, finding a significant $H_{c2}$ enhancement in C-alloyed films, and an anomalous upward curvature of $H_{c2}(T)$. Record high values of $H_{c2}^{perp}(4.2) simeq 35T$ and $H_{c2}|(4.2) simeq 51T$ were observed perpendicular and parallel to the ab plane, respectively. The temperature dependence of $H_{c2}(T)$ is described well by a theory of dirty two-gap superconductivity. Extrapolation of the experimental data to T=0 suggests that $H_{c2}|(0)$ approaches the paramagnetic limit of $sim 70T$.
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