Retroviral capsid proteins are conserved structurally but assemble into different morphologies. The mature human immunodeficiency virus-1 (HIV-1) capsid is best described by a 'fullerene cone' model, in which hexamers of the capsid protein are linked to form a hexagonal surface lattice that is closed by incorporating 12 capsid-protein pentamers. HIV-1 capsid protein contains an amino-terminal domain (NTD) comprising seven α-helices and a β-hairpin, a carboxy-terminal domain (CTD) comprising four α-helices, and a flexible linker with a 310-helix connecting the two structural domains. Structures of the capsid-protein assembly units have been determined by X-ray crystallography; however, structural information regarding the assembled capsid and the contacts between the assembly units is incomplete. Here we report the cryo-electron microscopy structure of a tubular HIV-1 capsid-protein assembly at 8 Å resolution and the three-dimensional structure of a native HIV-1 core by cryo-electron tomography. The structure of the tubular assembly shows, at the three-fold interface, a three-helix bundle with critical hydrophobic interactions. Mutagenesis studies confirm that hydrophobic residues in the centre of the three-helix bundle are crucial for capsid assembly and stability, and for viral infectivity. The cryo-electron-microscopy structures enable modelling by large-scale molecular dynamics simulation, resulting in all-atom models for the hexamer-of-hexamer and pentamer-of-hexamer elements as well as for the entire capsid. Incorporation of pentamers results in closer trimer contacts and induces acute surface curvature. The complete atomic HIV-1 capsid model provides a platform for further studies of capsid function and for targeted pharmacological intervention.