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 programme to search for companions around known extremely low-mass (ELM; ∼0.2 M⊙) WDs using the 6.5-m Multiple Mirror Telescope. We detect peak-to-peak radial velocity variations of 740 km s−1 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 2.1-m telescope reveal ellipsoidal variations from the distorted primary but no eclipses. This is the first example of a tidally distorted WD. Modelling the light curve, we constrain the inclination angle of the system to be 67°± 13°. J0106−1000 contains a pair of WDs (0.17 M⊙ primary + 0.43 M⊙ invisible secondary) at a separation of 0.32 R⊙. The two WDs will merge in 37 Myr and most likely form a core He-burning single subdwarf star. J0106−1000 is the shortest time-scale 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.