THE 'soft' spectra, short duration and recurrent activity of the three known soft γ-ray repeaters (SGRs) distinguishes them from the majority of cosmic γ-ray bursts1-4. Although there is no generally accepted model for the repeaters, it has been argued5 that they are associated with radio pulsars: this is consistent with the coincidence6of SGR0526 - 66 with the supernova remnant N49 in the Large Magellanic Cloud, and the location of the two other repeaters at low galactic latitudes. Here we show that the well localized repeater SGR1806 - 20 (refs 2,3) is also coincident with a supernova remnant: the amorphous radio nebula G 10.0 - 0.3 (refs 7,8). Taken together, these two associations argue strongly in favour of a neutron-star origin for the repeaters. We suggest that all young pulsars-that is, neutron stars still embedded in their supernova remnants-pass through a brief (~ 500 year) phase of SGR activity. The detection of pulsar-powered synchrotron nebulae in, or pulsations from, the two supernova remnants coincident with SGRs would confirm this model.