  We report the discovery of the first short period binary in which a hot
subdwarf star (sdOB) fills 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 $P_{\rm orb}=39.3401(1)$ min, 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 $T_{\rm eff}=42,400\pm300$ K and
a surface gravity of $\log(g)=5.77\pm0.05$. A high-signal-to noise 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 $M_{\rm sdOB}=0.337\pm0.015$ M$_\odot$ and a white dwarf
accretor with a mass $M_{\rm WD}=0.545\pm0.020$ M$_\odot$. Theoretical binary
modeling indicates the hot subdwarf formed during a common envelope phase when
a $2.5-2.8$ M$_\odot$ star lost its envelope when crossing the Hertzsprung Gap.
To match its current $P_{\rm orb}$, $T_{\rm eff}$, $\log(g)$, and masses, we
estimate a post-common envelope period of $P_{\rm orb}\approx150$ min, and find
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
$\approx0.1$ M$_\odot$ and will merge with its carbon/oxygen white dwarf
companion after $\approx17$ Myr and presumably explode as a thermonuclear
supernova or form an R CrB star.
