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&sun;) 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&sun; primary + 0.43 M&sun; invisible secondary) at a separation of 0.32 R&sun;. 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. Based on observations obtained at the Multiple Mirror Telescope (MMT) Observatory, a joint facility of the Smithsonian Institution and the University of Arizona.