The Immersion Grating Infrared Spectrometer (IGRINS) 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 (H2RG FPA) detectors. We present the design and fabrication of the detector mount for the H2RG detector. The detector mount consists of a detector housing, an ASIC housing, a Field Flattener Lens (FFL) mount, and a support base frame. The detector and the ASIC housing should be kept at 65 K and the support base frame at 130 K. Therefore they are thermally isolated by the support made of GFRP material. The detector mount is designed so that it has features of fine adjusting the position of the detector surface in the optical axis and of fine adjusting yaw and pitch angles in order to utilize as an optical system alignment compensator. We optimized the structural stability and thermal characteristics of the mount design using computer-aided 3D modeling and finite element analysis. Based on the structural and thermal analysis, the designed detector mount meets an optical stability tolerance and system thermal requirements. Actual detector mount fabricated based on the design has been installed into the IGRINS cryostat and successfully passed a vacuum test and a cold test.