We report the discovery of a remarkable ultra-compact dwarf galaxy around the massive Virgo elliptical galaxy NGC 4649 (M60), which we call M60-UCD1. With a dynamical mass of 2.0 × 108 M ☉ but a half-light radius of only ~24 pc, M60-UCD1 is more massive than any ultra-compact dwarfs of comparable size, and is arguably the densest galaxy known in the local universe. It has a two-component structure well fit by a sum of Sérsic functions, with an elliptical, compact (rh = 14 pc; n ~ 3.3) inner component and a round, exponential, extended (rh = 49 pc) outer component. Chandra data reveal a variable central X-ray source with LX ~ 1038 erg s–1 that could be an active galactic nucleus associated with a massive black hole or a low-mass X-ray binary. Analysis of optical spectroscopy shows the object to be old (gsim 10 Gyr) and of solar metallicity, with elevated [Mg/Fe] and strongly enhanced [N/Fe] that indicates light-element self-enrichment; such self-enrichment may be generically present in dense stellar systems. The velocity dispersion (σ ~ 70 km s–1) and resulting dynamical mass-to-light ratio (M/LV = 4.9 ± 0.7) are consistent with—but slightly higher than—expectations for an old, metal-rich stellar population with a Kroupa initial mass function. The presence of a massive black hole or a mild increase in low-mass stars or stellar remnants is therefore also consistent with this M/LV . The stellar density of the galaxy is so high that no dynamical signature of dark matter is expected. However, the properties of M60-UCD1 suggest an origin in the tidal stripping of a nucleated galaxy with MB ~ –18 to –19.