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We present a new method to measure the radius-to-mass ratio (R/M) of weakly magnetic, disk-accreting neutron stars by exploiting the occultation of parts of the inner disk by the star itself. This occultation imprints characteristic features on the X-ray line profile that are unique and are expected to be present in low-mass X-ray binary systems seen under inclinations higher than 65 degrees. We analyze a Nuclear Spectroscopic Telescope Array observation of a good candidate system, 4U 1636-53, and find that X-ray spectra from current instrumentation are unlikely to single out the occultation features owing to insufficient signal-to-noise. Based on an extensive set of simulations we show that large-area X-ray detectors of the future generation could measure R/M to 2 3% precision over a range of inclinations. Such is the precision in radius determination required to derive tight constraints on the equation of state of ultradense matter and it represents the goal that other methods also aim to achieve in the future.