A sufficiently high magnetic field applied to a superconductor will act on both the charge and the spin of the individual electrons breaking up the Cooper pairs. If the spin effect dominates, the superconducting state can develop a texture before eventually entering the normal state with increasing magnetic field. This spatially varying superconducting state is a periodic array of magnetic walls separated by superconducting regions. Known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, it was predicted in 1964. We report heat capacity measurements on the heavy fermion superconductor, CeCoIn5, which reveal a second phase transition within the superconducting state, clear evidence of the FFLO phase. We also report magnetization measurements that display a cascade of first order phase transitions within the FFLO region. Each transition indicates an increase in orbital momentum of the superconducting order parameter and corresponds to a specific Landau level vortex state comprised of multiquanta vortices. The experimental realization of the FFLO state provides a new opportunity to study the symbiosis of magnetism and superconductivity, two states of matter once thought to be mutually exclusive.