I report the discovery of blueshifted broad absorption line (BAL) troughs in at least six transitions of the Balmer series of hydrogen (Hβ to H9), and in Ca II, Mg II, and excited Fe II in the quasar SDSS J125942.80+121312.6. This is only the fourth active galactic nucleus known to exhibit Balmer absorption, all four of which also exhibit low-ionization BAL systems containing excited Fe II. The substantial population in the n = 2 shell of H I in this quasar's absorber likely arises from Lyα trapping. In an absorber sufficiently optically thick to show Balmer absorption, soft X-rays from the quasar penetrate to large τLyα and ionize H I. Recombination then creates Lyα photons that increase the n = 2 population by a factor of τLyα, since they require τLyα scatterings to diffuse out of the absorber. Observing Lyα trapping in a quasar absorber requires a large but Compton-thin column of gas along our line of sight, which includes substantial H I but not too much dust. Presumably, the rarity of Balmer line BAL troughs reflects the rarity of such conditions in quasar absorbers.