  We investigated the magnetic field dependence of the superconducting phase
transition in heavy fermion CeCoIn_5 (T_c = 2.3 K) using specific heat,
magneto-caloric effect, and thermal expansion measurements. The superconducting
transition becomes first order when the magnetic field is oriented along the
001 crystallographic direction with a strength greater that 4.7 T, and
transition temperature below T_0 ~ 0.31 T_c. The change from second order at
lower fields is reflected in strong sharpening of both specific heat and
thermal expansion anomalies associated with the phase transition, a strong
magnetocaloric effect, and a step-like change in the sample volume. The first
order superconducting phase transition in CeCoIn_5 is caused by Pauli limiting
in type-II superconductors, and was predicted theoretically in the mid 1960s.
We do not see evidence for the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) superconducting state (predicted by an alternative theory also dating
back to mid-60's) in CeCoIn_5 with field H || [001].
