  We investigate the acceleration of shock waves to relativistic velocities in
the outer layers of exploding stars. By concentrating the explosion energy in
the outermost ejecta, such trans-relativistic blast waves can serve as the
progenitors of gamma-ray bursts (GRBs); in particular, the ``baryon-loading''
problem that plagues many models of GRBs is circumvented. We present physically
motivated and numerically validated analytic expressions to describe
trans-relativistic blast waves in supernovae. We find that relativistic ejecta
are enhanced in more centrally condensed envelopes, e.g., for radiative
envelopes, when the luminosity approaches the Eddington limit. We present
convenient formulae for estimating the relativistic ejecta from a given
progenitor. We apply our analytic and numerical methods to a model of SN
1998bw, finding significantly enhanced relativistic ejecta compared to previous
studies. We propose that GRB 980425 is associated with SN 1998bw and may have
resulted from an approximately spherical explosion producing ~10^-6 M_sun of
mildly relativistic ejecta with mean Lorentz factor ~2, which then interacted
with a dense circumstellar wind with mass loss rate ~few x 10^-4 M_sun/yr. A
highly asymmetric explosion is not required. An extreme model of ``hypernova''
explosions in massive stars is able to account for the energetics and
relativistic ejecta velocities required by many of the observed cosmological
GRBs. We present simplified models of explosions resulting from
accretion-induced collapse of white dwarfs and phase transitions of neutron
stars. While we find increased energies in relativistic ejecta compared to
previous studies, these explosions are unlikely to be observed at cosmological
distances with current detectors. (abridged)
