Using the Hubble Space Telescope Goddard High Resolution Spectrograph (GHRS), we have obtained high S/N echelle observations of the weak interstellar N I λλ1160, 1161 absorption doublet toward the stars γ Cas, λ Ori, ι Ori, κ Ori, δ Sco, and κ Sco. In combination with a previous GHRS measurement of N I toward ζ Oph, these new observations yield a mean interstellar gas-phase nitrogen abundance (per 106 H atoms) of 106 N/H = 75 +/- 4 (+/-1 σ). There are no statistically significant variations in the measured N abundances from sight line to sight line and no evidence of density-dependent nitrogen depletion from the gas phase. Since N is not expected to be depleted much into dust grains in these diffuse sight lines, its gas-phase abundance should reflect the total interstellar abundance. Consequently, the GHRS observations imply that the abundance of interstellar nitrogen (gas plus grains) in the local Milky Way is about 80% of the solar system value of 106 N/H = 93 +/- 16. Although this interstellar abundance deficit is somewhat less than that recently found for oxygen and krypton with GHRS, the solar N abundance and the N I oscillator strengths are too uncertain to rule out definitively either a solar ISM N abundance or a 2/3 solar ISM N abundance similar to that of O and Kr.Based on observations obtained with the NASA/ESA Hubble Space Telescope through the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NASA-26555.