This paper proves the following converse to a theorem of Mumford: Let ; ; ; A; A; ; ; be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point is three. Then ; ; ; A; A; ; ; is isomorphic as a principally polarized abelian variety to the intermediate Jacobian of a smooth cubic threefold. The method of proof is to analyze the possible singularities of the theta divisor of ; ; ; A; A; ; ; , and ultimately to show that ; ; ; A; A; ; ; is the Prym variety of a possibly singular plane quintic.