White dwarfs are compact stars, similar in size to Earth but ~200,000 times more massive. Isolated white dwarfs emit most of their power from ultraviolet to near-infrared wavelengths, but when in close orbits with less dense stars, white dwarfs can strip material from their companions, and the resulting mass transfer can generate atomic line and X-ray emission, as well as near- and mid-infrared radiation if the white dwarf is magnetic. However, even in binaries, white dwarfs are rarely detected at far-infrared or radio frequencies. Here we report the discovery of a white dwarf / cool star binary that emits from X-ray to radio wavelengths. The star, AR Scorpii (henceforth AR Sco), was classified in the early 1970s as a delta-Scuti star, a common variety of periodic variable star. Our observations reveal instead a 3.56 hr period close binary, pulsing in brightness on a period of 1.97 min. The pulses are so intense that AR Sco's optical flux can increase by a factor of four within 30 s, and they are detectable at radio frequencies, the first such detection for any white dwarf system. They reflect the spin of a magnetic white dwarf which we find to be slowing down on a 10^7 yr timescale. The spin-down power is an order of magnitude larger than that seen in electromagnetic radiation, which, together with an absence of obvious signs of accretion, suggests that AR Sco is primarily spin-powered. Although the pulsations are driven by the white dwarf's spin, they originate in large part from the cool star. AR Sco's broad-band spectrum is characteristic of synchrotron radiation, requiring relativistic electrons. These must either originate from near the white dwarf or be generated in situ at the M star through direct interaction with the white dwarf's magnetosphere.