With a data sample containing 1.1×105 J/ψ→μ+μ− decays reconstructed with 16 MeV/c2 rms mass resolution, we have measured the differential cross sections versus Feynman-x, rapidity, and pT for the production of J/ψ and ψ’ in 800 GeV/c p-Au collisions. Our results are compared with leading-order QCD predictions and with previous measurements. While the shapes of the cross sections are in qualitative agreement with QCD predictions, the magnitudes disagree by factors of 7 (J/ψ) and 25 (ψ’). Assuming an appropriate form for the differential cross sections in regions not measured we derive a total J/ψ production cross section σ(p+N→J/ψ+X)=442±2±88 nb/nucleon and a (model-dependent) total ψ’ cross secton σ(p+N→ψ’+X)=75±5±22 nb/nucleon. For J/ψ produced at central rapidity, dσ(p+N→J/ψ+X)/dy‖y=0=230±5±46 nb/nucleon.