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 ∼0.034 ms, the gravitational-wave strain amplitude can increase by ∼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.