It was established over a decade ago that the remarkable high-energy transients known as soft gamma-ray repeaters (SGRs) are located in our Galaxy and originate from neutron stars with intense (< or = 10(15)G) magnetic fields-so-called 'magnetars'. On 27 December 2004, a giant flare with a fluence exceeding 0.3 erg cm(-2) was detected from SGR 1806-20. Here we report the detection of a fading radio counterpart to this event. We began a monitoring programme from 0.2 to 250 GHz and obtained a high-resolution 21-cm radio spectrum that traces the intervening interstellar neutral hydrogen clouds. Analysis of the spectrum yields the first direct distance measurement of SGR 1806-20: the source is located at a distance greater than 6.4 kpc and we argue that it is nearer than 9.8 kpc. If correct, our distance estimate lowers the total energy of the explosion and relaxes the demands on theoretical models. The energetics and the rapid decay of the radio source are not compatible with the afterglow model that is usually invoked for gamma-ray bursts. Instead, we suggest that the rapidly decaying radio emission arises from the debris ejected during the explosion.