Growing evidence suggests that the first generation of stars may have been quite massive (~100-300 M☉). If they retain their high mass until death, such stars will, after about 3 Myr, make pair-instability supernovae. Models for these explosions have been discussed in the literature for four decades, but very few included the effects of rotation and none employed a realistic model for neutrino trapping and transport. Both turn out to be very important, especially for those stars whose cores collapse into black holes (helium cores above about 133 M☉). We consider the complete evolution of two zero-metallicity stars of 250 and 300 M☉. Despite their large masses, we argue that the low metallicities of these stars imply negligible mass loss. Evolving the stars with no mass loss, but including angular momentum transport and rotationally induced mixing, the two stars produce helium cores of 130 and 180 M☉. Products of central helium burning (e.g., primary nitrogen) are mixed into the hydrogen envelope with dramatic effects on the radius, especially in the case of the 300 M☉ model. Explosive oxygen and silicon burning cause the 130 M☉ helium core (250 M☉ star) to explode, but explosive burning is unable to drive an explosion in the 180 M☉ helium core, and it collapses to a black hole. For this star, the calculated angular momentum in the presupernova model is sufficient to delay black hole formation, and the star initially forms an ~50 M☉, 1000 km core within which neutrinos are trapped. The calculated growth time for secular rotational instabilities in this core is shorter than the black hole formation time, and they may develop. If so, the estimated gravitational wave energy and wave amplitude are EGW ≈ 10-3 M☉ c2 and h+ ≈ 10-21/d(Gpc), but these estimates are very rough and depend sensitively on the nonlinear nature of the instabilities. After the black hole forms, accretion continues through a disk. The mass of the disk depends on the adopted viscosity but may be quite large, up to 30 M☉ when the black hole mass is 140 M☉. The accretion rate through the disk can be as large as 1-10 M☉ s-1. Although the disk is far too large and cool to transport energy efficiently to the rotational axis by neutrino annihilation, it has ample potential energy to produce a 1054 erg jet driven by magnetic fields. The interaction of this jet with surrounding circumstellar gas may produce an energetic gamma-ray transient, but given the probable redshift and the consequent timescale and spectrum, this model may have difficulty explaining typical gamma-ray bursts.