We report the discovery of the first short-period binary in which a hot subdwarf star (sdOB) filled its Roche lobe and started mass transfer to its companion. The object was discovered as part of a dedicated high-cadence survey of the Galactic plane named the Zwicky Transient Facility and exhibits a period of P = 39.3401(1) minutes, making it the most compact hot subdwarf binary currently known. Spectroscopic observations are consistent with an intermediate He-sdOB star with an effective temperature of Teff = 42,400 ± 300 K and a surface gravity of logg = 5.77 ± 0.05. A high signal-to-noise ratio GTC+HiPERCAM light curve is dominated by the ellipsoidal deformation of the sdOB star and an eclipse of the sdOB by an accretion disk. We infer a low-mass hot subdwarf donor with a mass M sdOB = 0.337 ± 0.015 M⊙ and a white dwarf accretor with a mass M WD = 0.545 ± 0.020 M⊙. Theoretical binary modeling indicates the hot subdwarf formed during a common envelope phase when a 2.5-2.8 M⊙ star lost its envelope when crossing the Hertzsprung gap. To match its current Porb, Teff, logg, and masses, we estimate a post-common envelope period of Porb ≈ 150 minutes and find that the sdOB star is currently undergoing hydrogen shell burning. We estimate that the hot subdwarf will become a white dwarf with a thick helium layer of ≈0.1 M⊙, merge with its carbon/oxygen white dwarf companion after ≈17 Myr, and presumably explode as a thermonuclear supernova or form an R CrB star.
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