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We present ultraviolet spectroscopy covering the Ly$α$ + N V complex of six candidate low-redshift ($0.9 < z < 1.5$) weak emission-line quasars (WLQs) based on observations with the Hubble Space Telescope. The original systematic searches for these puzzling Type 1 quasars with intrinsically weak broad emission lines revealed an $N \approx 100$ WLQ population from optical spectroscopy of high-redshift ($z > 3$) quasars, defined by a Ly$α$ + N V rest-frame equivalent width (EW) threshold $< 15.4$ Å. Identification of lower-redshift ($z < 3$) WLQ candidates, however, has relied primarily on optical spectroscopy of weak broad emission lines at longer rest-frame wavelengths. With these new observations expanding existing optical coverage into the ultraviolet, we explore unifying the low- and high-$z$ WLQ populations via EW[Ly$α$+NV]. Two objects in the sample unify with high-$z$ WLQs, three others appear consistent with the intermediate portion of the population connecting WLQs and normal quasars, and the final object is consistent with typical quasars. The expanded wavelength coverage improves the number of available line diagnostics for our individual targets, allowing a better understanding of the shapes of their ionizing continua. The ratio of EW[Ly$α$+NV] to EW[MgII] in our sample is generally small but varied, favoring a soft ionizing continuum scenario for WLQs, and we find a lack of correlation between EW[Ly$α$+NV] and the X-ray properties of our targets, consistent with a "slim-disk" shielding gas model. We also find indications that weak absorption may be a more significant contaminant in low-$z$ WLQ populations than previously thought.