We report on the discovery by the Swift Gamma-Ray Burst Explorer of the black hole (BH) candidate Swift J1539.2-6227 and the subsequent course of an outburst beginning in 2008 November and lasting at least seven months. The source was discovered during normal observations with the Swift Burst Alert Telescope on 2008 November 25. An extended observing campaign with the Rossi X-Ray Timing Explorer and Swift provided near-daily coverage over 176 days, giving us a good opportunity to track the evolution of spectral and timing parameters with fine temporal resolution through a series of spectral states. The source was first detected in a hard state during which strong low-frequency quasi-periodic oscillations (QPOs) were detected. The QPOs persisted for about 35 days and a signature of the transition from the hard- to soft-intermediate states was seen in the timing data. The source entered a short-lived thermal state about 40 days after the start of the outburst. There were variations in spectral hardness as the source flux declined and returned to a hard state at the end of the outburst. The progression of spectral states and the nature of the timing features provide strong evidence that Swift J1539.2-6227 is a candidate BH in a low-mass X-ray binary system.