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(1). Brown dwarfs older than similar to 10 Myr are expected to possess short-lived magnetic fields(2) and to emit radio and X-rays only very weakly from their coronae. An X-ray flare was recently detected(3) on the brown dwarf LP944-20, whereas previous searches(4-7) for optical activity (and one X-ray search(1)) 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(8,9) 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(10,11), which might explain the previous null optical(4-7) and X-ray(1) results, as well as the strength of the radio emissions compared to those at X-ray wavelengths.