We describe a double-scattering experiment with a novel tagged neutron beam to measure differential cross sections for np backscattering to better than +/-2% absolute precision. The measurement focuses on angles and energies where the cross section magnitude and angle dependence constrain the charged pion-nucleon coupling constant, but existing data show serious discrepancies among themselves and with energy-dependent partial-wave analyses. The present results are in good accord with the partial-wave analyses, but deviate systematically from other recent measurements.