The behavior of two electrons confined to a three-dimensional box with infinite walls and interacting with a Coulomb potential is studied using an exact diagonalization technique. The use of symmetry operators enables the Hamiltonian to be block diagonalized. Apart from the total spin, the wavefunctions can be classified using three symmetry quantum numbers. The Coulomb integrals are shown to be amenable to efficient and accurate calculation. The energy of the lowest few eigenstates of both the singlet (S=0) and triplet (S=1) are calculated as a function of the box size (i.e., in effect r(s)) for a slightly tetragonally distorted box where the z-axis is longer than the x- and y-axes. The ground state is a singlet function with ggg symmetry at all densities. At small r(s), the ground state has a maximum in electron density at the box center. Upon increasing r(s), at r(s)approximate to8 a.u., the ground state density acquires a minimum in the box center. At this same r(s), the first-excited state of the singlet manifold changes its symmetry from ggu to ugu, and the corresponding degeneracy is changed from one to two. The energy-r(s) curve shows a nonanalytic change in slope. Subsequent increasing of r(s) gives rise to increased localization of the charge density in the eight corners of the box, which can be identified as the "Wigner" crystal limit of the present model. The physical exchange-correlation hole is evaluated in the high and low density limits.