The pairing of fermions lies at the heart of superconductivity and superfluidity. The stability of these pairs determines the robustness of the superfluid state, and the quest for superconductors with high critical temperature equates to a search for systems with strong pairing mechanisms. Ultracold atomic Fermi gases present a highly controllable model system for studying strongly interacting fermions1. Tunable interactions (through Feshbach collisional resonances) and the control of population or mass imbalance among the spin components provide unique opportunities to investigate the stability of pairing2, 3, 4—and possibly to search for exotic forms of superfluidity5, 6. A major controversy has surrounded the stability of superfluidity against an imbalance between the two spin components when the fermions interact resonantly (that is, at unitarity). Here we present the phase diagram of a spin-polarized Fermi gas of 6Li atoms at unitarity, experimentally mapping out the superfluid phases versus temperature and density imbalance. Using tomographic techniques, we reveal spatial discontinuities in the spin polarization; this is the signature of a first-order superfluid-to-normal phase transition, and disappears at a tricritical point where the nature of the phase transition changes from first-order to second-order. At zero temperature, there is a quantum phase transition from a fully paired superfluid to a partially polarized normal gas. These observations and the implementation of an in situ ideal gas thermometer provide quantitative tests of theoretical calculations on the stability of resonant superfluidity.