We present a solid-state laser system that generates over 200mW of continuous-wave, narrowband light, tunable from 316.3 nm -317.7 nm and 318.0 nm - 319.3 nm. The laser is based on commercially available fiber amplifiers and optical frequency doubling technology, along with sum frequency generation in a periodically poled stoichiometric lithium tantalate crystal. The laser frequency is stabilized to an atomic-referenced high finesse optical transfer cavity. Using a GPS-referenced optical frequency comb we measure a long term frequency instability of <35 kHz for timescales between 10-3 s and 103 s. As an application we perform spectroscopy of Sr Rydberg states from n = 37 - 81, demonstrating mode-hopfree scans of 24 GHz. In a cold atomic sample we measure Doppler-limited linewidths of 350 kHz.