We present SDSS J092712.65+294344.0 as the best candidate to date for a recoiling supermassive black hole (SMBH). SDSS J0927+2943 shows an exceptional optical emission-line spectrum with two sets of emission lines: one set of very narrow emission lines, and a second set of broad Balmer and broad high-ionization forbidden lines which are blueshifted by 2650 km s⁻¹ relative to the set of narrow emission lines. This observation is most naturally explained if the SMBH was ejected from the core of the galaxy, carrying with it the broad-line gas while leaving behind the bulk of the narrow-line gas. We show that the observed properties of SDSS J0927+2943 are consistent with predictions and expectations from recent numerical relativity simulations which demonstrate that SMBHs can receive kicks up to several thousand km s⁻¹ due to anisotropic emission of gravitational waves during the coalescence of a binary. Our detection of a strong candidate for a rapidly recoiling SMBH implies that kicks large enough to remove SMBHs completely from their host galaxies do occur, with important implications for models of black hole and galaxy assembly at the epoch of structure formation, and for recoil models.