[en] We report constraints on the global 21 cm signal due to neutral hydrogen at redshifts $14.8⩾<!--\; ???\; -->z⩾<!--\; ???\; -->6.5$. We derive our constraints from low-foreground observations of the average sky brightness spectrum conducted with the EDGES High-band instrument between 2015 September 7 and October 26. Observations were calibrated by accounting for the effects of antenna beam chromaticity, antenna and ground losses, signal reflections, and receiver parameters. We evaluate the consistency between the spectrum and phenomenological models for the global 21 cm signal. For tanh-based representations of the ionization history during the epoch of reionization, we rule out, at $⩾<!--\; ???\; -->2\mathrm{\sigma <!--\; ??\; -->}$ significance, models with duration of up to $\mathrm{\Delta <!--\; ??\; -->}z=1$ at $z\approx <!--\; ???\; -->8.5$ and higher than $\mathrm{\Delta <!--\; ??\; -->}z=0.4$ across most of the observed redshift range under the usual assumption that the 21 cm spin temperature is much larger than the temperature of the cosmic microwave background during reionization. We also investigate a “cold” intergalactic medium (IGM) scenario that assumes perfect Lyα coupling of the 21 cm spin temperature to the temperature of the IGM, but that the latter is not heated by early stars or stellar remants. Under this assumption, we reject tanh-based reionization models of duration $\mathrm{\Delta <!--\; ??\; -->}z\lesssim <!--\; ???\; -->2$ over most of the observed redshift range. Finally, we explore and reject a broad range of Gaussian models for the 21 cm absorption feature expected in the First Light era. As an example, we reject 100 mK Gaussians with duration (full width at half maximum) $\mathrm{\Delta <!--\; ??\; -->}z⩽<!--\; ???\; -->4$ over the range $14.2⩾<!--\; ???\; -->z⩾<!--\; ???\; -->6.5$ at $⩾<!--\; ???\; -->2\mathrm{\sigma <!--\; ??\; -->}$ significance.