  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 $\sim 70T$.
