The type 1 human immunodeficiency virus (HIV-1) contains a conical capsid comprising approximately 1,500 CA protein subunits, which organizes the viral RNA genome for uncoating and replication in a new host cell. In vitro, CA spontaneously assembles into helical tubes and cones that resemble authentic viral capsids. Here we describe electron cryo-microscopy and image reconstructions of CA tubes from six different helical families. In spite of their polymorphism, all tubes are composed of hexameric rings of CA arranged with approximate local p6 lattice symmetry. Crystal structures of the two CA domains were 'docked' into the reconstructed density, which showed that the amino-terminal domains form the hexameric rings and the carboxy-terminal dimerization domains connect each ring to six neighbours. We propose a molecular model for the HIV-1 capsid that follows the principles of a fullerene cone, in which the body of the cone is composed of curved hexagonal arrays of CA rings and the ends are closed by inclusion of 12 pentagonal 'defects'.