For the second time in 27 years a bright transient X-ray source has been detected coincident with the globular cluster NGC 6440. It was found to be active in August, 1998, with the Wide Field Camera and the narrow field instruments on the BeppoSAX spacecraft, and with the All-Sky Monitor and the Proportional Counter Array on the RossiXTE spacecraft. Four X-ray bursts were detected, at least one of which shows the characteristics of a thermonuclear flash on a neutron star, in analogy with some ~20 optically identified low-mass X-ray binaries. The broad-band spectrum is hard as is common among low-mass X-ray binaries of lower luminosity (>~10^37 erg/s) and can be explained by a Comptonized model. During the burst the >30 keV emission brightened, consistent with part of the burst emission being Compton up scattered within ~10^11 cm.