This paper describes a high dispersion spectrograph (HDS) now under construction towards the first-light phase of Subaru Telescope. It is located at a Nasmyth focus. HDS is an echelle spectrograph with grating as a post-disperser. The collimated beam size is 272 mm, and the echelle is 300 mm by 840 mm in total size (31.6 gr/mm, R-2.8). The overall throughput (resolution X slit width product) achieved is 38,000 arcseconds. HDS has two cross-dispersers (400 gr/mm, 250 gr/mm) which are optimized for blue- and red-wavelength regions, respectively. The camera is of catadioptric type system, consisting of three correctors and a mirror. It spans the entire chromatic range from 0.30 to 2.0 micrometer. It delivers 10-micrometer images on average within a flat 60 mm- diameter focal plane area in the wavelength range from 0.3 to 1.1 micrometer, without refocusing. This image quality corresponds to typical limiting spectral resolutions well above 300,000 though the resolution will generally be limited to less than this by the entrance slit width and finite CCD pixel sizes. The detector will be a 2 X 1 mosaic of 2k X 4k CCD's with 15-micrometer pixels. Typical limiting spectral resolution will be 100,000.