Many sub-stellar companions (usually planets but also some brown dwarfs) orbit solar-type stars. These stars can engulf their sub-stellar companions when they become red giants. This interaction may explain several outstanding problems in astrophysics but it is unclear under what conditions a low mass companion will evaporate, survive the interaction unchanged or gain mass. Observational tests of models for this interaction have been hampered by a lack of positively identified remnants-that is, white dwarf stars with close, sub-stellar companions. The companion to the pre-white dwarf AA Doradus may be a brown dwarf, but the uncertain history of this star and the extreme luminosity difference between the components make it difficult to interpret the observations or to put strong constraints on the models. The magnetic white dwarf SDSS J121209.31 + 013627.7 may have a close brown dwarf companion but little is known about this binary at present. Here we report the discovery of a brown dwarf in a short period orbit around a white dwarf. The properties of both stars in this binary can be directly observed and show that the brown dwarf was engulfed by a red giant but that this had little effect on it.