We report the discovery of an object near M87 in the Virgo Cluster with an extraordinary blueshift of -1025 km s-1, offset from the systemic velocity by >2300 km s-1. Evaluation of photometric and spectroscopic data provides strong evidence that this object is a distant massive globular cluster, which we call HVGC-1 in analogy to Galactic hypervelocity stars. We consider but disfavor more exotic interpretations, such as a system of stars bound to a recoiling black hole. The odds of observing an outlier as extreme as HVGC-1 in a virialized distribution of intracluster objects are small; it appears more likely that the cluster was (or is being) ejected from Virgo following a three-body interaction. The nature of the interaction is unclear, and could involve either a subhalo or a binary supermassive black hole at the center of M87.