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Superintegrable Hamiltonians in three degrees of freedom possess more than three functionally independent globally defined and single-valued integrals of motion. Some familiar examples, such as the Kepler problem and the harmonic oscillator, have been known since the time of Laplace. Here, a classification theorem is given for superintegrable potentials with invariants that are quadratic polynomials in the canonical momenta. Such systems must possess separable solutions to the Hamilton-Jacobi equation in more than one coordinate system. There are 11 coordinate systems for which the Hamilton-Jacobi equation separates in ${\mathit{openR}}^{3}$. One coordinate system may be arbitrarily rotated or translated with respect to the other, yielding 66 distinct cases. In each case, the differential equations for separability in the two coordinates are integrated to give a complete list of all superintegrable potentials with four or five quadratic integrals. The tables---which may be consulted independently of the main body of the paper---list the distinct superintegrable potentials, the separating coordinates, and the isolating integrals of the motion. If there exist five isolating integrals, then all finite classical trajectories are closed; if only four, then the trajectories are restricted to two-dimensional surface. An extraordinary consequence of the work is the discovery of perturbations to both the Kepler problem and the harmonic oscillator that do not destroy the fragile degeneracy. The perturbed systems still have five isolating integrals of the motion.