We have performed full-scale numerical-renormalization-group calculations on the anisotropic, two-channel, impurity spin s= 1/2 Kondo Hamiltonian. To allow for anisotropy we have worked in a basis of total axial charge j a and j b (a and b index the channels) as well as total S z. In the isotropic limit, our results agree well with the previous work of Cragg, Lloyd, and Nozieres. The most important results are (1) confirmation of the approach to a nontrivial fixed point from weak coupling,(2) the stability of the fixed point against exchange anisotropy, and (3) the destabilization of the fixed point by the application of a magnetic field or the lifting of exchange symmetry between two channels. We find the crossover temperature scales with the square of the exchange splitting.