IGRINS, the Immersion GRating INfrared Spectrometer, is a near-infrared wide-band high-resolution spectrograph jointly developed by the Korea Astronomy and Space Science Institute and the University of Texas at Austin. IGRINS employs three HAWAII-2RG focal plane array (FPA) detectors. The mechanical mounts for these detectors serves a critical function in the overall instrument design: Optically, they permit the only positional compensation in the otherwise "build to print" design. Thermally, they permit setting and control of the detector operating temperature independently of the cryostat bench. We present the design and fabrication of the mechanical mount as a single module. The detector mount includes the array housing, a housing for the SIDECAR ASIC, a field flattener lens holder, and a support base. The detector and ASIC housing will be kept at 65 K and the support base at 130 K. G10 supports thermally isolate the detector and ASIC housing from the support base. The field flattening lens holder attaches directly to the FPA array housing and holds the lens with a six-point kinematic mount. Fine adjustment features permit changes in axial position and in yaw and pitch angles. We optimized the structural stability and thermal characteristics of the mount design using computer-aided 3D modeling and finite element analysis. Based on the computer simulation, the designed detector mount meets the optical and thermal requirements very well.