  We report the results of a new analysis of the ROSAT Wide Field Camera (WFC)
all-sky survey data, designed to detect extragalactic sources of extreme
ultraviolet (EUV) radiation in regions of low Galactic N_H. We identify a total
of 19 active galactic nuclei (AGN), more than double the number of confirmed
AGN in the published WFC (2RE) survey. Our sample contains 8 narrow-line
Seyfert 1 galaxies, making this the first reasonably-sized, complete sample of
these extreme AGN, along with 6 broad-line Seyfert 1s and 5 BL Lacertae
objects. These EUV-selected sources typically have steep soft X-ray spectra
with a mean power-law energy index alpha_X=-2. The derived AGN luminosity
function is rather flat and appears to cut off above a luminosity of 10^46
erg/s/keV, measured monochromatically at 200 eV. Narrow-line Seyfert~1s account
for roughly half of the local (z~0) volume emissivity in the EUV band.
