Nonblinking excitonic emission from near-infrared and type-II nanocrystal quantum dots (NQDs) is reported for the first time. To realize this unusual degree of stability at the single-dot level, novel InP/CdS core/shell NQDs were synthesized for a range of shell thicknesses (~1-11 monolayers of CdS). Ensemble spectroscopy measurements (photoluminescence peak position and radiative lifetimes) and electronic structure calculations established the transition from type-I to type-II band alignment in these heterostructured NQDs. More significantly, single-NQD studies revealed clear evidence for blinking suppression that was not strongly shell-thickness dependent, while photobleaching and biexciton lifetimes trended explicitly with extent of shelling. Specifically, very long biexciton lifetimes-up to >7 ns-were obtained for the thickest-shell structures, indicating dramatic suppression of nonradiative Auger recombination. This new system demonstrates that electronic structure and shell thickness can be employed together to effect control over key single-dot and ensemble NQD photophysical properties.