  We report first-principles calculations of the superconducting properties of
$\beta$--Bi$_2$Pd within the fully anisotropic Migdal-Eliashberg formalism. We
find a single anisotropic superconducting gap of $s$--wave symmetry which
varies in magnitude on the different regions of the Fermi surface. The
calculated superconducting energy gap on the Fermi surface, the superconducting
transition temperature, the specific heat, and the quasi-particle density of
states are in good agreement with the corresponding experimental results and
support the view that $\beta$--Bi$_2$Pd is a phonon-mediated single anisotropic
gap superconductor.
