We report multi-epoch VLA H I absorption observations of the source 1741-038 (OT-068) before and during an extreme scattering event (ESE). Observations at four epochs, three during the ESE, were obtained. We find no changes in the equivalent width, maximum optical depth, or velocity of maximum optical depth during the ESE, but we do find a secular trend of decreasing maximum optical depth between our observations and ones by other observers a decade prior. The resulting limit on the H I column density change during the ESE for a structure with a spin temperature T_s is 6.4 x 10^{17} cm^{-2} (T_s/10 K). Tiny-scale atomic structures (TSAS), with a column density N_H ~ 3 x 10^{18} cm^{-2}, are ruled out marginally by this limit, though geometric arguments may allow this limit to be relaxed. Galactic halo molecular clouds, that are opaque in the H I line, cannot be excluded because the observed velocity range covers only 25% of their allowed velocity range.
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