We report on the first observation of a pronounced reentrant superconductivity phenomenon in a superconductor/ferromagnet layered system. The results were obtained using a superconductor/ferromagnetic-alloy bilayer of Nb/Cu(1-x)Ni(x). The superconducting transition temperature T(c) drops sharply with increasing thickness dCuNi) of the ferromagnetic layer, until complete suppression of superconductivity is observed at d(CuNi) approximately equal to 4 nm. Increasing the Cu(1-x)Ni(x) layer thickness further, superconductivity reappears at d(CuNi) > or =13 nm. Our experiments give evidence for the pairing function oscillations associated with a realization of the quasi-one-dimensional Fulde-Ferrell-Larkin-Ovchinnikov-like state in the ferromagnetic layer.