We report multi-epoch Very Large Array 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 Ðnd no changes in the equivalent width, maximum optical depth, or velocity of maximum optical depth during the ESE, but we do Ðnd 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 is 6.4 ] 1017 T s K). Tiny-scale atomic structures (TSAS), with cm~2, are ruled out marginally cm~2(T s /10 N H D 3 ] 1018 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 from causing the ESE because the observed velocity range covers only 25% of their allowed velocity range.