We report the observation of a nonlinear propagation scenario in which a dispersive wave is transformed into a fundamental soliton in an axially varying optical fiber. The dispersive wave is initially emitted in the normal dispersion region and the fiber properties change longitudinally so that the dispersion becomes anomalous at the dispersive wave wavelength, which allows it to be transformed into a soliton. The solitonic nature of the field is demonstrated by solving the direct Zakharov-Shabat scattering problem. Experimental characterization performed in spectral and temporal domains show evidence of the solitonization process in an axially varying photonic crystal fiber.