Using new and archival radio data, we have measured the proper motion of the black hole X-ray binary V404 Cyg to be 9.2+/-0.3 mas/yr. Combined with the systemic radial velocity from the literature, we derive the full three-dimensional heliocentric space velocity of the system, which we use to calculate a peculiar velocity in the range 47-102 km/s, with a best fitting value of 64 km/s. We consider possible explanations for the observed peculiar velocity, and find that the black hole cannot have formed via direct collapse. A natal supernova is required, in which either significant mass (approximately 11 solar masses) was lost, giving rise to a symmetric Blaauw kick of up to 65 km/s, or, more probably, asymmetries in the supernova led to an additional kick out of the orbital plane of the binary system. In the case of a purely symmetric kick, the black hole must have been formed with a mass of approximately 9 solar masses, since when it has accreted 0.5-1.5 solar masses from its companion.
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