  Radio-frequency spectroscopy is used 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 is
observed. This demonstrates that 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.
