Active galactic nuclei, which are powered by long-term accretion onto central supermassive black holes, produce relativistic jets with lifetimes of at least one million years, and the observation of the birth of such a jet is therefore unlikely. Transient accretion onto a supermassive black hole, for example through the tidal disruption of a stray star, thus offers a rare opportunity to study the birth of a relativistic jet. On 25 March 2011, an unusual transient source (Swift J164449.3+573451) was found, potentially representing such an accretion event. Here we report observations spanning centimetre to millimetre wavelengths and covering the first month of evolution of a luminous radio transient associated with Swift J164449.3+573451. The radio transient coincides with the nucleus of an inactive galaxy. We conclude that we are seeing a newly formed relativistic outflow, launched by transient accretion onto a million-solar-mass black hole. A relativistic outflow is not predicted in this situation, but we show that the tidal disruption of a star naturally explains the observed high-energy properties and radio luminosity and the inferred rate of such events. The weaker beaming in the radio-frequency spectrum relative to γ-rays or X-rays suggests that radio searches may uncover similar events out to redshifts of z~6.