We report a major outburst from the anomalous X-ray pulsar 1E 2259+586, in which over 80 X-ray bursts were detected in 4 hr using the Rossi X-Ray Timing Explorer. The bursts range in duration from 2 ms to 3 s and have fluences in the 2-10 keV band that range from 3 × 10-11 to 5 × 10-9 ergs cm-2. We simultaneously observed increases of the pulsed and persistent X-ray emission by over an order of magnitude relative to quiescent levels. Both decayed significantly during the course of our 14 ks observation. Correlated spectral hardening was also observed, with the spectrum softening during the observation. In addition, we observed a pulse profile change, in which the amplitudes of the two peaks in the pulse profile were swapped. The profile relaxed back to its pre-outburst morphology after ~6 days. The pulsar also underwent a sudden spin-up (Δν/ν = 4 × 10-6), followed by a large (factor of ~2) increase in spin-down rate that persisted for more than 18 days. We also observed, using the Gemini North telescope, an infrared enhancement, in which the Ks (2.15 μm) flux increased, relative to that measured in a observation made in 2000, by a factor of ~3, 3 days post-outburst. The IR counterpart then faded by a factor of ~2 1 week later. In addition, we report an upper limit of 50 μJy on radio emission at 1.4 GHz 2 days post-outburst. The X-ray properties of this outburst are like those seen only in soft gamma repeaters. This conclusively unifies anomalous X-ray pulsars and soft gamma repeaters, as predicted uniquely by the magnetar model.