  The anomalous Hall effect is investigated experimentally and theoretically
for ferromagnetic thin films of Mn$_5$Ge$_3$. We have separated the intrinsic
and extrinsic contributions to the experimental anomalous Hall effect, and
calculated the intrinsic anomalous Hall conductivity from the Berry curvature
of the Bloch states using first-principles methods. The intrinsic anomalous
Hall conductivity depends linearly on the magnetization, which can be
understood from the long wavelength fluctuations of the spin orientation at
finite temperatures. The quantitative agreement between theory and experiment
is remarkably good, not only near 0 K, but also at finite temperatures, up to
about ~ 240 K (0.8 T$_C$})
