We have measured the neutron-proton total cross section in hydrogen gas at 0°C between 0.3- and 400-eV lab energy to help improve the data for computing the n-p singlet effective range rs. To obtain the free n-p scattering cross section at "zero" energy, we subtract the capture-cross-section, effective-range, and molecular-binding contributions from the total cross section. We used the asymptotic "MTV" theory of molecular binding due to Messiah, which limits our final analysis to the range 6.02 to 329 eV. From these data we obtain 20.436 +/- 0.023 b for the free n-p scattering cross section. Using this number, the coherent-scattering-length measurements by Koester, a model due to Lomon and Feshbach for the triplet and singlet shape parameters, and the n-p cross section in the MeV range due to Engelke et al., we obtain rs=2.56+/-0.05 F, 0.09 F higher than the former value. Adopting an energy scale due to Davis and Barschall for the higher-energy data raises rs to 2.74 F, which is in close agreement with the value 2.70 F predicted by the charge-independence hypothesis. A measurement of the neutron-carbon cross section, formerly related to this problem, is also discussed.