We are developing a High Resolution Spectrograph (HRS) for ultraviolet astronomy with the Space Telescope. The instrument will provide a spectral resolution of ≈ 1.2 x 105 over a nominal wavelength range of 110 - 320 nm, together with a spatial resolution of about 0.25 arc seconds. The two detectors will consist of 512-element Digicons with cesium telluride and cesium iodide photocathodes, respectively. Photoelectrons in transit between the photocathodes and the diodes within the Digicons can be deflected in two axes with 12-bit resolution. This feature facilitates a design that emphasizes reliability since (once a hermetic seal is opened in orbit), only two moving parts, a grating carrousel and a shutter, are required for regular operation of the HRS. The instrument will be controlled by a computer in the spacecraft. The scientific objectives of the HRS investigation relate to interstellar matter in our own and nearby galaxies, physical processes of stellar mass loss and mass transfer, chemical abundances, bright quasars and Seyfert galaxy nuclei, and solar system phenomena.