Kidney exchange creates gains by pooling patient-donor pairs across hospitals and countries, but cooperation may unravel when coalitions can profitably withdraw. We introduce the supplemented core, in which the platform uses voluntarily registered altruistic donors to restore stability. Because an altruistic donor adds a kidney without adding another patient, it provides an in-kind instrument for relaxing participation constraints when monetary transfers are unavailable. In worst-case compatibility graphs, the required number of donors can grow linearly with market size, even when cycle length is unrestricted. Under a standard heterogeneous random-compatibility model, however, a logarithmic number suffices for any fixed cycle-length bound, in expectation and with high probability. Calibrated simulations find donor-free, maximum-cardinality weak-core exchanges in virtually all markets. The main challenge is selection, not existence: a representative lexicographic heuristic reflecting priorities used by kidney-exchange programs selects an unstable exchange in up to 35% of markets even when a stable alternative exists. A small reserve of altruistic donors eliminates this implementation gap without sacrificing the heuristic's operational objectives. Thus, altruistic donors do more than increase transplants: they sustain cooperation.

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