The observations of CO and SiO in the infrared spectrum of SN 1987A clearly indicate that molecules can form in the debris of a supernova explosion. Since H2 is not easily observable we compute its abundance theoretically. For conditions typical of the inner (υ < 2500 km s-1) envelope of SN 1987A, the fraction of H that is in molecular form rises to ∼1% by t ∼ 800 days. For t < 500 days the formation is dominated by the gas-phase reactions H + H+ → H+2 + hv; H+2 + H → H2 + H+. Thereafter, the formation is dominated by the reactions H + e → H- + hv; H- + H → H2 + e. At early times the H- may absorb ∼10%-30% of visible photons, contributing to the apparent paucity of Hα emission. For t > 1000 days the abundance of H2 "freezes out" due to the slowing of all reactions. The opacity of the supernova envelope in the range 912 < λ ≤ 1400 Å (the upper limit depending on temperature) is dominated by resonance scattering in the Lyman and Werner bands of H2. The resulting fluorescence emission bands of H2 in the range 1150 < λ < 1650 Å may be observable in the UV spectra of supernovae at late times.