The divergent parts of the five two-loop quark self-energy diagrams of quantum chromodynamics are evaluated in the noncovariant 'light-cone gauge'. Most of the Feynman integrals are computed by means of the powerful 'matrix integration method', originally developed for the author's Master's thesis. From the results of the integrations, it is shown how to renormalize the quark mass and wave function in such a way that the effective quark propagator is rendered finite at two-loop order. The required counterterms turn out to be 'local' functions of the quark momentum, due to cancellation of the nonlocal divergent parts of the two-loop integrals with equal and opposite contributions from one-loop counterterm subtraction diagrams. The final form of the counterterms is seen to be consistent with the renormalization framework proposed by Bassetto, Dalbosco, and Soldati, in which all noncovariant divergences are absorbed into the wave function normalizations. It also turns out that the mass renormalization d 'm' is the same in the light-cone gauge as it is in a general 'covariant' gauge, at least up to two-loop order.