From its creation in 1989 through subsequent extensions, the widely-used "SnapPea census" now aims to represent all cusped finite-volume hyperbolic 3-manifolds that can be obtained from <= 8 ideal tetrahedra. Its construction, however, has relied on inexact computations and some unproven (though reasonable) assumptions, and so its completeness was never guaranteed. For the first time, we prove here that the census meets its aim: we rigorously certify that every ideal 3-manifold triangulation with <= 8 tetrahedra is either (i) homeomorphic to one of the census manifolds, or (ii) non-hyperbolic. In addition, we extend the census to 9 tetrahedra, and likewise prove this to be complete. We also present the first list of all minimal triangulations of all census manifolds, including non-geometric as well as geometric triangulations.