  We report an enhancement of thermal conductivity ($\kappa$) below the
superconducting transition temperature ($T_C$) in the high carrier density
$\beta$-V$_{1-x}$Ti$_x$ alloys. We find that the point defects generated when
Ti is added to V reduce the electron mean free path down to the inter-atomic
distances and make the high frequency phonons ineffective in carrying heat. In
this Mott-Ioffe-Regel limit, the phonon thermal conductivity is dominated by
the low frequency phonons limited by the scattering due to the electrons. The
formation of Cooper pairs below the $T_C$ re-normalizes the phonon mean free
path and enhances the $\kappa$.
