  Although gravitational waves only interact weakly with matter, their
propagation is affected by a gravitational potential. If a gravitational wave
source is eclipsed by a star, measuring these perturbations provides a way to
directly measure the distribution of mass throughout the stellar interior. We
compute the expected Shapiro time delay, amplification and deflection during an
eclipse, and show how this can be used to infer the mass distribution of the
eclipsing body. We identify continuous gravitational waves from neutron stars
as the best candidates to detect this effect. When the Sun eclipses a far-away
source, depending on the depth of the eclipse the time-delay can change by up
to $\sim 0.034$ ms, the gravitational-wave strain amplitude can increase by
$\sim 4$%, and the apparent position of the source in the sky can vary by
$4''$. Accreting neutron stars with Roche-lobe filling companion stars have a
high probability of exhibiting eclipses, producing similar time delays but
undetectable changes in amplitude and sky location. Even for the most rapidly
rotating neutron stars, this time delay only corresponds to a few percent of
the phase of the gravitational wave, making it an extremely challenging
measurement. However, if sources of continuous gravitational waves exist just
below the limit of detection of current observatories, next-generation
instruments will be able to observe them with enough precision to measure the
signal of an eclipsing star. Detecting this effect would provide a new direct
probe to the interior of stars, complementing asteroseismology and the
detection of solar neutrinos.
