  We present numerical simulations of primordial supernovae in cosmological
minihalos at $z \sim$ 20. We consider Type II supernovae, hypernovae, and pair
instability supernovae (PISN) in halos from 6.9 $\times$ 10$^5$ - 1.2 $\times$
10$^7$ $\Ms$, those in which Population III stars are expected to form via
H$_2$ cooling. The supernovae evolve along two evolutionary paths according to
whether they explode in \ion{H}{2} regions or neutral halos. Those in
\ion{H}{2} regions first expand adiabatically and then radiate strongly upon
collision with baryons ejected from the halo during its photoevaporation by the
progenitor. Explosions in neutral halos promptly emit most of their kinetic
energy as x-rays, but retain enough momentum to seriously disrupt the halo. We
find that the least energetic of the supernovae are capable of destroying halos
$\lesssim$ 10$^7$ $\Ms$, while a single PISN can destroy even more massive
halos. Blasts in \ion{H}{2} regions disperse heavy elements into the IGM, but
neutral halos confine the explosion and its metals. In \ion{H}{2} regions, a
prompt second generation of stars may form in the remnant at radii of 100 - 200
pc in the halo. Explosions confined by large halos instead recollapse, with
infall rates in excess of 10$^{-2}$ $\Ms$ yr$^{-1}$ that heavily contaminate
their interior. This fallback may either fuel massive black hole growth at very
high redshifts or create the first globular cluster with a radius of 10 - 20 pc
at the center of the halo. Our findings allow the possibility that the first
primitive galaxies formed sooner, with greater numbers of stars and distinct
chemical abundance patterns, than in current models.
