In recent years, millisecond duration radio signals originating from distant galaxies appear to have been discovered in the so-called Fast Radio Bursts. These signals are dispersed according to a precise physical law and this dispersion is a key observable quantity which, in tandem with a redshift measurement, can be used for fundamental physical investigations. While every fast radio burst has a dispersion measurement, none before now have had a redshift measurement, due to the difficulty in pinpointing their celestial coordinates. Here we present the discovery of a fast radio burst and the identification of a fading radio transient lasting $sim 6$ days after the event, which we use to identify the host galaxy; we measure the galaxys redshift to be $z=0.492pm0.008$. The dispersion measure and redshift, in combination, provide a direct measurement of the cosmic density of ionised baryons in the intergalactic medium of $Omega_{mathrm{IGM}}=4.9 pm 1.3%$, in agreement with the expectation from WMAP, and including all of the so-called missing baryons. The $sim6$-day transient is largely consistent with a short gamma-ray burst radio afterglow, and its existence and timescale do not support progenitor models such as giant pulses from pulsars, and supernovae. This contrasts with the interpretation of another recently discovered fast radio burst, suggesting there are at least two classes of bursts.
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