The Hubble Space Telescope 2000-8000 Angstrom spectrum of SN 1987A observed on 1995 January 7 (7.87 yr after the explosion) is dominated by H alpha and UV lines, including Mg n-Mg I lambda 2825 (equal to H alpha in luminosity), Fe II UV 2 (two-thirds the intensity of H alpha), Fe II UV 3 (one-half the intensity of H alpha), and a 3730 Angstrom emission feature identified with a blend of [O II] lambda 3727 and Fe I emission lines. [O I] lambda 6300 and lines of [Ca II] and Na I, as well as some Fe II optical forbidden and permitted lines are present at visual wavelengths. Also present are a number of weak emission features, which are presumably metal lines produced by photon degradation as a result of reprocessing of UV radiation into metal lines. Modeling the Mg II-Mg I lines provides the velocity of the outer visible radius of the envelope, 9000 +/- 500 km s(-1) in the Mg II lambda 2800 line, which is consistent with the earlier direct HST imaging at near-UV wavelengths. The UV/optical emission lines originate from the radioactive luminescence of the cool gas (T approximate to 130-160 K). The metal lines reflect the instantaneous reprocessing of the energy deposited from Ti-44 radioactive decays through collisions with fast electrons, while the H alpha emission primarily comes from the recombination of previously ionized hydrogen. The overall luminosity of the Fe II emission lines, similar to 10(35) ergs s(-1), can be explained if the bulk of the positrons from a mass (1-2) x 10(-4) M. of Ti-44 release their energy in the iron-rich material, which suggests the presence of a magnetic field B > 5 x 10(-13) G prohibiting the escape of positrons into oxygen and hydrogen gas. The ionized fraction in the iron-rich material is small (0.2-0.3), and the total UV/optical emission from Fe I should be comparable to that from Fe II. Most of the 10(36) ergs s(-1) deposited by the Ti-44 positions Should be emitted in the Fe II 26 mu m Line. The observed H alpha luminosity decrease, by 5 orders of magnitude between the ages of 1 to 8 yr, is reproduced in a time-dependent model of ionization and cooling with the ''standard'' amount of radioactive nuclides. However, an additional source of energy at the present epoch with a deposition rate 30 ergs s(-1) g(-1) (approximate to 10(36) ergs s(-1) in the whole envelope) is not ruled out. The present average temperature in the hydrogen envelope predicted by the time-dependent model is 130 K, which is lower than the value T approximate to 350 K obtained from the observed Balmer continuum shape. However, the shape is affected by a possible contribution of metal lines to the Balmer continuum. The luminosity of the [O I] lambda 6300 doublet is consistent with that expected for the deposited energy of gamma-rays from (1-2) x 10(-4) M. of Ti-44 for an for the amount of Ti-44 in SN 1987A. The maximum fraction of the Ti-44 positron energy deposited into oxygen-rich material does not exceed 5%, which is consistent with positron trapping in Fe-rich material. The [O I] lambda 6300 line intensity rules out the presence of a central source of gamma-radiation (hv > 100 keV) with a luminosity L-gamma > 4 x 10(36) ergs s(-1).