Elastic neutron-scattering, inelastic x-ray scattering, specific-heat, and pressure-dependent electrical transport measurements have been made on single crystals of AuZn and Au0.52Zn0.48. Elastic neutron scattering detects new commensurate Bragg peaks (modulation) appearing at Q =(1.33,0.67,0) at temperatures corresponding to each sample's transition temperature (TM = 64 and 45 K, respectively). Although the new Bragg peaks appear in a discontinuous manner in the Au0.52Zn0.48 sample, they appear in a continuous manner in AuZn. Surprising us, the temperature dependence of the AuZn Bragg peak intensity and the specific-heat jump near TM are in favorable accord with a continuous transition. A fit to the pressure dependence of TM suggests the presence of a critical end point in the AuZn phase diagram located at TM* = 2.7 K and p* = 3.1 GPa.