JWST, despite not being designed to observe astrophysical phenomena that vary on rapid timescales, can be an unparalleled tool for such studies. If timing systematics can be controlled, JWST will be able to open up the subsecond infrared timescale regime. Rapid time-domain studies, such as lag measurements in accreting compact objects and solar system stellar occultations, require both precise interframe timing and knowing when a time series begins, down to an absolute accuracy significantly below 1 s. In this work, we present two long-duration observations of the deeply eclipsing double white dwarf system ZTF J153932.16+502738.8, which we use as a natural timing calibrator to measure the absolute timing accuracy of JWST's clock. From our two epochs, we measure an average clock accuracy of 0.12 ± 0.06 s, implying that JWST can be used for subsecond time-resolution studies down to the ∼100 ms level, a factor ∼5 improvement upon the prelaunch clock accuracy requirement. We also find an asymmetric eclipse profile in the F322W2 band, which we suggest has a physical origin.