We report multi-epoch Very Large Array H I absorption observations of the s
ource 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 othe
r observers a decade prior. The resulting limit on the H I column density c
hange during the ESE for a structure with a spin temperature T-s is 6.4 x 1
0(17) cm(-2)(T-s/10 K). Tiny-scale atomic structures (TSAS), with N-H simil
ar to 3 x 10(18) cm(-2), are ruled out marginally by this limit, though geo
metric arguments may allow this limit to be relaxed. Galactic halo molecula
r clouds that are opaque in the H I line cannot be excluded from causing th
e ESE because the observed velocity range covers only 25% of their allowed
velocity range.