An examination of data from the Apollo-Soyuz diffuse background survey and of earlier observations obtained with rocket-borne proportional counters has provided new constraints on the cosmic EUV spectrum in the 20--280 eV range. This spectrum is analyzed using a hot interstellar plasma emission model wherein the plasma is in steady state ionization equilibrium and is either isothermal or characterized by a differential emission measure with a power-law temperature distribution. In both cases, if the emitting plasma is behind a slab of absorbing material of column density N/sub H/, the EUV data is compatible with models having T<10/sup 5.8/ K and N/sub H/< or =10/sup 19.5/ cm/sup -2/. If the absorbing gas is interspersed with the plasma, only isothermal models with 10/sup 5.3/< or =T< or =10/sup 5.5/ K yield satisfactory fits to the observations. The allowed model parameters are consistent with the predictions of the McKee and Ostriker theory of the interstellar medium but inconsistent with the observed parameters of the O VI absorbing gas. Reconciliation with the O VI observations in the isothermal case is achieved only if the gas is not in pressure equilibrium with its surroundings or if most of the observed background is due to unresolvedmore » point sources. Temperature distribution models allow marginal agreement between theory and observations only if the local O VI density is 4a4 times higher than the average over larger distances for a gas in pressure equilibrium. Alternately, the data may be taken as evidence of the necessity of discarding the steady state ionization equilibrium condition in favor of time-dependent models.« less