We used radio-frequency spectroscopy to study pairing in the normal and superfluid phases of a strongly interacting Fermi gas with imbalanced spin populations. At high spin imbalances, the system does not become superfluid even at zero temperature. In this normal phase, full pairing of the minority atoms was observed. Hence, mismatched Fermi surfaces do not prevent pairing but can quench the superfluid state, thus realizing a system of fermion pairs that do not condense even at the lowest temperature. Fermionic superfluidity has many manifes-tations in nature; it occurs in such diversesystems as superconducting materials, liquid 3He, neutron stars, and ultracold quantum gases. At its heart lies the formation of fermion pairs. Although the P...