The specific heat of the layered organic superconductor $kappa$-% (BEDT-TTF)$_2$Cu(NCS)$_2$, where BEDT-TTF is bisethylenedithio-% tetrathiafulvalene, has been studied in magnetic fields up to 28 T applied perpendicular and parallel to the superconducting layers. In parallel fields above 21 T, the superconducting transition becomes first order, which signals that the Pauli-limiting field is reached. Instead of saturating at this field value, the upper critical field increases sharply and a second first-order transition line appears within the superconducting phase. Our results give strong evidence that the phase, which separates the homogeneous superconducting state from the normal state is a realization of a Fulde-Ferrell-Larkin-Ovchinnikov state.
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