We have discovered a new transient low-mass X-ray binary, NGC 6440 X-2, with Chandra/ACIS, RXTE/PCA, and Swift/XRT observations of the globular cluster NGC 6440. The discovery outburst (2009 July 28-31) peaked at LX ~ 1.5 × 1036 erg s-1 and lasted for <4 days above LX = 1035 erg s-1. Four other outbursts (2009 May 29-June 4, August 29-September 1, October 1-3, and October 28-31) have been observed with RXTE/PCA (identifying millisecond pulsations) and Swift/XRT (confirming a positional association with NGC 6440 X-2), with similar peak luminosities and decay times. Optical and infrared imaging did not detect a clear counterpart, with best limits of V>21, B>22 in quiescence from archival Hubble Space Telescope imaging, g'>22 during the August outburst from Gemini-South GMOS imaging, and J >~ 18.5 and K >~ 17 during the July outburst from CTIO 4 m ISPI imaging. Archival Chandra X-ray images of the core do not detect the quiescent counterpart (LX < (1-2) × 1031 erg s-1) and place a bolometric luminosity limit of L NS < 6 × 1031 erg s-1 (one of the lowest measured) for a hydrogen atmosphere neutron star. A short Chandra observation 10 days into quiescence found two photons at NGC 6440 X-2's position, suggesting enhanced quiescent emission at LX ~ 6 × 1031 erg s-1. NGC 6440 X-2 currently shows the shortest recurrence time (~31 days) of any known X-ray transient, although regular outbursts were not visible in the bulge scans before early 2009. Fast, low-luminosity transients like NGC 6440 X-2 may be easily missed by current X-ray monitoring.