  We report first-principles calculations of the electronic band structure and
lattice dynamics for the new superconductor MgB$_{2}$. The excellent agreement
between theory and our inelastic neutron scattering measurements of the phonon
density of states gives confidence that the calculations provide a sound
description of the physical properties of the system. The numerical results
reveal that the in-plane boron phonons (with E$_{2g}$ symmetry) near the
zone-center are very anharmonic, and are strongly coupled to the partially
occupied planar B $\sigma$ bands near the Fermi level. This giant anharmonicity
and non-linear electron-phonon coupling is key to explaining the observed high
T$_{c}$ and boron isotope effect in MgB$_{2}$
