Infectious HIV particles contain a characteristic cone-shaped core encasing the viral RNA and replication proteins. The core exhibits significant heterogeneity in size and shape, yet consistently forms a well-defined structure. The mechanism by which the core is assembled in the maturing virion remains poorly understood. Using cryo-electron tomography, we have produced three-dimensional reconstructions of authentic, unstained HIV-1. These reveal the viral morphology with unprecedented clarity and suggest the following mechanism for core formation inside the extracellular virion: core growth initiates at the narrow end of the cone and proceeds toward the distal side of the virion until limited by the viral membrane. Curvature and closure of the broad end of the core are then directed by the inner surface of the viral membrane. This mechanism accommodates significant flexibility in lattice growth while ensuring the closure of cores of variable size and shape.