We have investigated the temperature and magnetic field dependence of the Hall coefficient of two well-characterized superconducting MgB$_2$ films (T$_{c0}$=38.0 K) in both the normal and superconducting states. Our results show that the normal-state Hall coefficient R$_H$ is positive and increases with decreasing temperature, independent of the applied magnetic field. Below T$_c$(H), R$_H$ decreases rapidly with temperature and changes sign before it reaches zero. The position and magnitude at which R$_H$ shows a minimum depends on the applied field. Quantitative analysis of our data indicates that the Hall response of MgB$_2$ behaves very similarly to that of high-T$_c$ cuprates: R$_H$ $propto$ T and cot$theta_H$ $propto$ T$^2$ in the normal state, and a sign reversal of R$_H$ in the mixed state. This suggests that the B-B layers in MgB$_2$, like the Cu-O planes in high-T$_c$ cuprates, play an important role in the electrical transport properties.
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