Despite their technological importance, lead sulfide (PbS) nanocrystals have lagged behind nanocrystals of cadmium selenide (CdSe) and lead selenide (PbSe) in terms of size and energy homogeneity. Here, we show that the ratio of lead to sulfur precursor available during nucleation is a critical parameter affecting subsequent growth and monodispersity of PbS nanocrystal ensembles. Applying this knowledge, we synthesize highly monodisperse (size dispersity <5%) PbS nanocrystals over a wide range of sizes (exciton energies from 0.70 to 1.25 eV, or 1000-1800 nm) without the use of size-selective precipitations. This degree of monodispersity results in absorption peak half width at half max (HWHM) values as small as 20 meV, indicating an ensemble that is close to the homogeneous limit. Photoluminescence emission is correspondingly narrow and exhibits small Stokes shifts and quantum efficiencies of 30-60%. The nanocrystals readily self-assemble into ordered superlattices and exhibit exceptional air stability over several months.