  The rest-frame ultraviolet (UV) spectra of active galactic nuclei (AGNs) are
important diagnostics of both accretion disk physics and their contribution to
the metagalactic ionizing UV background. Though the mean AGN spectrum is well
characterized with composite spectra at wavelengths greater than 912 Angstroms,
the shorter-wavelength extreme-UV (EUV) remains poorly studied. In this third
paper in a series on the spectra of AGNs, we combine 11 new spectra taken with
the Cosmic Origins Spectrograph on the Hubble Space Telescope with archival
spectra to characterize the typical EUV spectral slope of AGNs from
$\lambda_{\rm rest}\sim 850~{\rm Angstroms}$ down to $\lambda_{\rm rest}\sim
425~{\rm Angstroms}$. Parameterizing this slope as a power law, we obtain
$F_\nu\propto \nu^{ -0.72\pm 0.26}$, but we also discuss the limitations and
systematic uncertainties of this model. We identify broad emission features in
this spectral region, including emission due to ions of O, Ne, Mg, and other
species, and we limit the intrinsic HeI 504 Angstrom photoelectric absorption
edge opacity to $\tau_{\rm HeI}<0.047$.
