The existence of small-scale variations in the interstellar medium, between sight lines separated by 10-100 AU, is well established from pulsar, VLBI, MERLIN, and VLA observations. The angular resolutions are in the range 10-100 mas. During an outburst of the microquasar GRS 1915+105, the proper motion of the approaching component is similar to 250 AU (23 mas) day(-1). Using the VLA to observe the H I line at 21 cm, we have measured the time-variable H I absorption along the changing line of sight during such an outburst. We detect opacity changes of Delta tau similar to 0.67 +/- 0.16 in gas at upsilon(LSR) of 55 km s(-1), near the velocity of the tangent point at 65 km s(-1), over similar to 6 days, for lines of sight similar to 1000 AU apart. Another detection at upsilon(LSR) of 5 km s(-1) has Delta tau similar to 0.24 +/- 0.06, on lines of sight similar to 40 AU (or 440 AU) apart, depending on the near (or far) distance to the absorbing gas. From the several days of spectra, 3 sigma upper limits are derived: Delta tau less than or equal to 0.5 for gas at 55 km s(-1), on scales similar to 150-900 AU; and Delta tau less than or equal to 0.25 at 5 km s(-1), on scales similar to 25-150 AU (gas at 1 kpc) or similar to 275-1650 AU (gas at 11 kpc). We thus probe the H I structure in distant Galactic material, using a Galactic background source with rapid structural changes. Our approach complements those using extragalactic sources and nearby pulsars. Our results are consistent with extrapolation of the power-law index of 0.375 for the structure function of H I opacity (equivalent to an index of 2.75 for the power spectrum of H I), as measured by others from 4 pc to 0.02 pc toward Cassiopeia A.