So far, nonsequential double ionization (NSDI) of atoms can be well understood within a semiclassical or even classical picture. No quantum effect appears to be required to explain the data observed. We theoretically study electron correlation resulting from NSDI of argon in a low-intensity laser field using a quantum-mechanical S-matrix theory. We show that quantum interference between the contributions of different intermediate excited states of the singly charged argon ion produces a transition from back-to-back to side-by-side emission with increasing laser intensity, which is in close agreement with the experimental data. For higher intensities, this transition is enhanced by the consequences of depletion of the excited states.