Model many-body wave functions are constructed for a quantized vortex line and ring. The corresponding variational energies are calculated using an approximate integral equation derived by Percus and Yevick for classical fluids. The energy of the line is close to that found in the Hartree theory. However, the density distribution in the core region is markedly different, and the core size is somewhat smaller, being of the order of 1 Å. The velocity of translation of the vortex ring is calculated by a new method, and excellent agreement is obtained with the experimental results of Rayfield and Reif. The condensed-state wave function is computed numerically, and it is found that in the core of the vortex there are about 20% fewer particles in the condensed state than in the zero-momentum state when the system is in equilibrium.