We identify SDSS J010657.39-100003.3 (hereafter J0106-1000) as the shortest
period detached binary white dwarf (WD) system currently known. We targeted
J0106-1000 as part of our radial velocity program to search for companions
around known extremely low-mass (ELM, ~ 0.2 Msol) WDs using the 6.5m MMT. We
detect peak-to-peak radial velocity variations of 740 km/s with an orbital
period of 39.1 min. The mass function and optical photometry rule out a
main-sequence star companion. Follow-up high-speed photometric observations
obtained at the McDonald 2.1m telescope reveal ellipsoidal variations from the
distorted primary but no eclipses. This is the first example of a tidally
distorted WD. Modeling the lightcurve, we constrain the inclination angle of
the system to be 67 +- 13 deg. J0106-1000 contains a pair of WDs (0.17 Msol
primary + 0.43 Msol invisible secondary) at a separation of 0.32 Rsol. The two
WDs will merge in 37 Myr and most likely form a core He-burning single subdwarf
star. J0106-1000 is the shortest timescale merger system currently known. The
gravitational wave strain from J0106-1000 is at the detection limit of the
Laser Interferometer Space Antenna (LISA). However, accurate ephemeris and
orbital period measurements may enable LISA to detect J0106-1000 above the
Galactic background noise.Comment: MNRAS Letters, in pres
