  (Abridged) We report multi-epoch VLBI observations of the source PKS 1741-038
as it underwent an extreme scattering event. Observations at four epochs were
obtained, and images were produced at three of these. During the event the
source consisted of a dominant, compact component, essentially identical to the
structure seen outside the event. However, the source's diameter increased
slightly at 13 cm during the ESE. An increase in the source's diameter is
inconsistent with a simple refractive model. We also see no evidence for
ESE-induced substructure within the source or the formation of multiple images,
as would occur in a strongly refractive lens. However, a model in which the
decrease in flux density during the ESE occurs solely because of stochastic
broadening within the lens requires a larger broadening diameter during the ESE
than is observed. Thus, the ESE toward 1741-038 involved both stochastic
broadening and refractive defocussing within the lens. If the structure
responsible for the ESE has a size of order 1 AU, the level of scattering
within an ESE lens may be a factor of 10^7 larger than that in the ambient
medium. A filamentary structure could reduce the difference between the
strength of scattering in the lens and ambient medium, but we conclude that, if
ESEs arise from filamentary structures, they occur when the filamentary
structures are seen lengthwise. We predict the amount of pulse broadening that
would result from a comparable lens passing in front of a pulsar. The pulse
broadening would be no more than 1.1 microseconds, consistent with the lack of
pulse broadening detected during ESEs toward the pulsars PSR B1937+21 and PSR
J1643-1224.
