Abstract: Fermionic superfluidity with mismatched Fermi surfaces has been an intriguing problem for many decades. Cooper pairing is the underlying mechanism for the conventional Bardeen-Cooper-Schrieffer (BCS) superfluid state of an equal, two component Fermi mixture. A mismatch between the Fermi surfaces of the two components, however strains the fully-paired state and eventually leads to the breakdown of superfluidity. This is known as the Chandrasekhar-Clogston (CC) limit of superfluidity. We have experimentally investigated the superfluid and normal phases of a strongly interacting Fermi gas with imbalanced spin populations. As the population imbalances was increased the Fermi mixture maintained superfluidity up to a critical imbalance, where the CC limit of superfluidity was reached. A novel phase-contrast imaging technique was employed to determine the spatial structure of the imbalanced Fermi mixture and the superfluid and the normal component were found to phase separate: The superfluid region of equal spin densities spatially separated from a normal region with unequal spin densities. Radio-frequency spectroscopy was used to study pairing in the strongly interacting normal phase. Surprisingly, full pairing of the minority component was observed even above the CC limit. This demonstrates that mismatched Fermi surfaces do not prevent pairing but quench the superfluid state, thus realizing a system of fermion pairs that do not condense even at the lowest temperature.