Measurements have been made of ACTT for polarized neutrons incident on a polarized proton target from 3.65 to 11.60 MeV. In the energy range near 10 MeV, ACTT is very sensitive to the nucleon-nucleon tensor interaction. Comparison of the data to potential-model predictions indicate that the tensor interaction is weak, resulting in values of the ^Si-^Di mixing parameter €i which are smaller than predicted by any nucleon-nucleon potential model. A smaller tensor force will bring the predictions of local potential models for the triton binding energy into closer agreement with the experimental value.PACS numbers: 25.40.Dn, 13.75.Cs, 21.30.+y, 25.10.+S A long-standing problem in nuclear physics has been the theoretical prediction of the triton binding energy [1]. All realistic and local nucleon-nucleon (NN) potentials predict too little binding. Many mechanisms have been proposed to increase the triton binding energy, among which are three-body forces and relativistic effects. Recent theoretical work, however, suggests that neither mechanism can contribute significantly [2,3]. In contrast, it is well established that the strength of the NN tensor force below 50 MeV has a large influence on the triton binding energy [4]. A^A^ parameters above 50 MeV are not important for this problem as this is approximately the Fermi energy of the nucleons in the triton. Calculations [5] show that a weak tensor force at low energies will bring potential model predictions into closer agreement with the experimentally determined value.In spite of its importance, both in the binding of fewnucleon systems and in the saturation of nuclear matter [5], the strength of the NN tensor force is only loosely constrained by the existing data [6]. We have measured the spin-dependent difference in total cross section, Aor, for the scattering of transversely polarized neutrons from transversely polarized protons below 12 MeV. These measurements cover a significant fraction of the energy range important to the triton binding energy. Our measurements indicate that the tensor force is indeed weak at low energies, possibly resolving the triton binding energy problem.At low energies, the strength of the tensor interaction is parametrized by the isoscalar ^Si-^Di phase-shift mixing parameter ei. Aar exhibits a large sensitivity to ei for neutron energies from 5 to 35 MeV [7]. In addition, ACT is insensitive to most other phase-shift parameters. The only significant sensitivity is to the ^SQ and ^Si phase shifts which are constrained by experiment. In contrast, other observables sensitive to the tensor interaction such as the spin transfer parameter, K^ , and the spin-correlation coefficient, Ayy{9)^ are sensitive to the P waves, which are much less well determined. Only in the special case 9 = 90^ is Ayy independent of the -^Pi phase shift.ACT is measured using a polarized neutron beam incident upon a polarized proton target and is defined to be the total cross section with the spins antiparallel minus the total cross section with the spins parallel [8]:A...