We present an analysis from simultaneous FUSE, ASCA, and EUVE observations, as well as a reanalysis of archival HST spectra, from the extreme narrow-line Seyfert 1 Galaxy RE 1034+39 (KUG 1031+398). RE 1034+39 has an unusually hard spectral energy distribution ( SED) that peaks in the soft X-rays. Its emission lines are unusual in that they can all be modeled as a Lorentzian centered at the rest wavelength with only a small range in velocity widths. In order to investigate whether the unusual SED influences the emission-line ratios and equivalent widths, we present three complementary types of photoionization analysis. The FUSE spectrum was particularly important because it includes the high-ionization line O VI. First, we use the photoionization code CLOUDY and the SED developed from the coordinated observations to confirm that the emission lines are consistent with observed hard SED. The best model parameters were an ionization parameter log U approximate to 2 and a hydrogen number density log n(H) = 9.75 (cm(-2)). Second, we present a locally optimally emitting cloud model. This model produced enhanced O vi as observed, but also yielded far too strong Mg II. Third, we develop a series of semiempirical SEDs, run CLOUDY models, and compare the results with the measured values using a figure of merit (FOM). The FOM minimum indicates SED and gas properties similar to those inferred from the one-zone model using the RE 1034+39 continuum. Furthermore, the FOM increases sharply toward softer continua, indicating that a hard SED is required by the data in the context of a one-zone model.