  We solve the coupled gap equations for the $\sigma$- and $\pi$-bands of
MgB$_2$ in the vortex state and calculate the resulting field dependencies of
the specific heat coefficient $\gamma$ and the thermal conductivity $\kappa $.
The crucial parameters of the theory are the interband pairing interaction
$\lambda_{\pi\sigma}$ and the ratio $s=\xi_{\sigma}/\xi_{\pi}$ of the coherence
lengths. For reasonably small $\lambda_{\pi\sigma}$ and s, the small gap
$\Delta_{\pi}$ decreases with increasing magnetic field $H$ much faster than
the large gap $\Delta_{\sigma} $. This gives rise to the observed rapid
increase of $\gamma_{\pi}$ and $\kappa_{\pi}$ for small fields while
$\gamma_{\sigma}$ and $\kappa_{\sigma}$ exhibit conventional field
dependencies. Inclusion of intraband impurity scattering yields fairly good
agreement with experiments for applied fields along the c axis.
