Using astrometric VLBI observations, we have determined the parallax of the black hole X-ray binary V404 Cyg to be 0.418 +- 0.024 mas, corresponding to a distance of 2.39 +- 0.14 kpc, significantly lower than the previously accepted value. This model-independent estimate is the most accurate distance to a Galactic stellar-mass black hole measured to date. With this new distance, we confirm that the source was not super-Eddington during its 1989 outburst. The fitted distance and proper motion imply that the black hole in this system likely formed in a supernova, with the peculiar velocity being consistent with a recoil (Blaauw) kick. The size of the quiescent jets inferred to exist in this system is <1.4 AU at 22 GHz. Astrometric observations of a larger sample of such systems would provide useful insights into the formation and properties of accreting stellar-mass black holes.