We report on polarized beam{endash}polarized target measurements of the spin-dependent neutron-proton total cross-section differences in longitudinal and transverse geometries ({Delta}{sigma}{sub L} and {Delta}{sigma}{sub T} , respectively) between E{sub n}=5 and 20thinspthinspMeV. Single-parameter phase-shift analyses were performed to extract the phase-shift mixing parameter {var_epsilon}{sub 1} , which characterizes the strength of the nucleon-nucleon tensor interaction at low energies. Consistent with the trend of previous determinations at E{sub n}=25 and 50thinspthinspMeV, our values for {var_epsilon}{sub 1} imply a stronger tensor force than predicted by meson-exchange nucleon-nucleon potential models and nucleon-nucleon phase-shift analyses. {copyright} {ital 1999} {ital The American Physical Society}