The photon total cross section on protons has been measured with high precision in the Fermilab tagged-photon beam for photon energies from 18 to 185 GeV. The cross section decreases to a broad minimum near 40 GeV, and then rises by about 4 μb over the remainder of the range. A ρ+ω+ϕ vector-dominance model (normalized to low-energy data) falls below the high-energy results by 2 to 6 μb, suggesting a contribution from charm-anticharm states.