We construct a flux-limited sample of 135 candidate z~1 Ly{alpha} emitters (LAEs) from Galaxy Evolution Explorer (GALEX) grism data using a new data cube search method. These LAEs have luminosities comparable to those at high redshifts and lie within a 7Gyr gap present in existing LAE samples. We use archival and newly obtained optical spectra to verify the UV redshifts of these LAEs. We use the combination of the GALEX UV spectra, optical spectra, and X-ray imaging data to estimate the active galactic nucleus (AGN) fraction and its dependence on Ly{alpha} luminosity. We remove the AGNs and compute the luminosity function (LF) from 60 z~1 LAE galaxies. We find that the best-fit LF implies a luminosity density increase by a factor of ~1.5 from z~0.3 to z~1 and ~20 from z~1 to z~2. We find a z~1 volumetric Ly{alpha} escape fraction of 0.7%+/-0.4%.