The most plausible cosmological model dominated by baryonic dark matter, characterized by primeval isocurvature fluctuations, may be ruled out in the limit of very early reionization by the new COBE FIRAS limit y < 2.5 x 10(-5) on Compton y-distortions of the cosmic microwave background. A neutral phase is inevitable for such a model to survive this confrontation with data. Indeed, not only this model, but virtually any model with similar cosmological parameters (OMEGA0 = OMEGA(b) almost-equal-to 0.15, OMEGA(igm) almost-equal-to 0.04, h almost-equal-to 0.8), in which the intergalactic medium is ionized as early as z = 800 is ruled out. This conclusion is independent of whether the main ionization mechanism is collisional ionization due to high temperatures or photoionization at lower temperatures. Completely ionized scenarios with typical cold dark matter parameters (OMEGA = 1, h2OMEGA(b) almost-equal-to 0.015) are still allowed for most realistic spectra of ionizing radiation, but will also be ruled out if the gamma-limit were to be improved by another factor of a few. If the ionizing radiation has a typical quasar spectrum, then the gamma-constraint implies roughly h3/2OMEGA(igm)OMEGA0(-1/4) < 0.025 for fully ionized models.