  We report the first magneto-caloric and calorimetric observations of a
magnetic-field-induced phase transition within a superconducting state to the
long-sought exotic "FFLO" superconducting state first predicted over 50 years
ago. Through the combination of bulk thermodynamic calorimetric and
magnetocaloric measurements in the organic superconductor $\kappa$ -
(BEDT-TTF)$_2$Cu(NCS)$_2$, as a function of temperature, magnetic field
strength, and magnetic field orientation, we establish for the first time that
this field-induced first-order phase transition at the paramagnetic limit $H_p$
for traditional superconductivity is to a higher entropy superconducting phase
uniquely characteristic of the FFLO state. We also establish that this
high-field superconducting state displays the bulk paramagnetic ordering of
spin domains required of the FFLO state. These results rule out the alternate
possibility of spin-density wave (SDW) ordering in the high field
superconducting phase. The phase diagram determined from our measurements ---
including the observation of a phase transition into the FFLO phase at $H_p$
--- is in good agreement with recent NMR results and our own earlier
tunnel-diode magnetic penetration depth experiments, but is in disagreement
with the only previous calorimetric report.
