The nucleus of the Milky Way is known to harbour regions of intense star formation activity as well as a supermassive black hole. Recent observations have revealed regions of γ-ray emission reaching far above and below the Galactic Centre (relative to the Galactic plane), the so-called 'Fermi bubbles'. It is uncertain whether these were generated by nuclear star formation or by quasar-like outbursts of the central black hole and no information on the structures' magnetic field has been reported. Here we report observations of two giant, linearly polarized radio lobes, containing three ridge-like substructures, emanating from the Galactic Centre. The lobes each extend about 60 degrees in the Galactic bulge, closely corresponding to the Fermi bubbles, and are permeated by strong magnetic fields of up to 15 microgauss. We conclude that the radio lobes originate in a biconical, star-formation-driven (rather than black-hole-driven) outflow from the Galaxy's central 200 parsecs that transports a huge amount of magnetic energy, about 10 55 ergs, into the Galactic halo. The ridges wind around this outflow and, we suggest, constitute a 'phonographic' record of nuclear star formation activity over at least ten million years.