To search for low-redshift damped Lyα (DLA) and sub-DLA quasar absorbers, we have conducted a 21 cm absorption survey of radio-loud quasars at small impact parameters to foreground galaxies selected from the Sloan Digital Sky Survey (SDSS). Here we present the first results from this survey based on observations of SDSS J104257.58+074850.5 (z QSO = 2.66521), a quasar at an angular separation from a foreground galaxy (z gal = 0.03321) of 25 (1.7 kpc in projection). The foreground galaxy is a low-luminosity spiral with on-going star formation (0.004 M yr–1 kpc–2) and a metallicity of –0.27 ± 0.05 dex. We detect 21 cm absorption from the galaxy with the Green Bank Telescope (GBT), the Very Large Array (VLA), and the Very Long Baseline Array (VLBA). The absorption appears to be quiescent disk gas co-rotating with the galaxy and we do not find any evidence for outflowing cold neutral gas. The width of the main absorption line indicates that the gas is cold, Tk < 283 K, and the H I column is surprisingly low given the impact parameter of 1.7 kpc; we find that N(H I) ≤9.6 × 1019 cm–2 (GBT) and N(H I) ≤1.5 × 1020 cm–2 (VLBA). VLBA marginally resolves the continuum source and the absorber, and a lower limit of 27.1 × 13.9 pc is derived for the size of the absorbing cloud. In turn, this indicates a low density for a cold cloud, n(H I) < 3.5 cm–3. We hypothesize that this galaxy, which is relatively isolated, is becoming depleted in H I because it is converting its interstellar matter into stars without a replenishing source of gas, and we suggest future observations to probe this and similar galaxies.