The penetrable sphere fluid consists of a system of spherical particles interacting via a potential that remains finite and constant for distances smaller than the particle diameter and is zero otherwise. This system, which was proposed sometime ago as a model for micelles in a solvent, has represented so far a remarkable challenge for integral equation theories which proved unable to correctly model the behavior of the two-body correlations inside the particle overlap region. It is shown in this work that enforcing the fulfillment of zero separation theorems for the cavity distribution function y(r), and thermodynamic consistency conditions (fluctuation vs virial compressibility and Gibbs-Duhem relation), on a parametrized closure of the type proposed by Verlet, leads to an excellent agreement with simulation, both for the thermodynamics and the structure (inside and outside the particle core). Additionally, the behavior of the integral equation at high packing fractions is explored and the bridge functions extracted from simulation are compared with the predictions of the proposed integral equation.