Brown dwarfs are not massive enough to sustain thermonuclear fusion of hydrogen at their centres, but are distinguished from gas-giant planets by their ability to burn deuterium. Brown dwarfs older than ~10Myr are expected to possess short-lived magnetic fields and to emit radio and X-rays only very weakly from their coronae. An X-ray flare was recently detected on the brown dwarf LP944-20, whereas previous searches for optical activity (and one X-ray search) yielded negative results. Here we report the discovery of quiescent and flaring radio emission from LP944-20, with luminosities several orders of magnitude larger than predicted by the empirical relation between the X-ray and radio luminosities that has been found for many types of stars. Interpreting the radio data within the context of synchrotron emission, we show that LP944-20 has an unusually weak magnetic field in comparison to active M-dwarf stars, which might explain the previous null optical and X-ray results, as well as the strength of the radio emissions compared to those at X-ray wavelengths.