We have developed a theory for self-diffusion in single-file Langmuirian zeolites of finite extent, which has been validated by open system kinetic Monte Carlo simulations. Our theory is based on a two-stage, Fickian diffusion mechanism, wherein a vacancy must traverse the entire file length to produce particle displacements of one lattice spacing. For times shorter than the vacancy diffusion time, t(c), particle transport proceeds via the nonFickian, single-file diffusion mode, with mean-square displacements increasing with the square-root of time. For times longer than t(c), however, we find that self-diffusion in single-file systems is completely described by Fick's laws. We find that the fraction of time in the single-file diffusion mode scales inversely with file length for long files, suggesting that Fickian self-diffusion dominates transport in longer single-file zeolites. Through correlations among the particle movements, the single-file self-diffusivity is sensitive to sorption limitations for short files, and scales inversely with file length for long files. Experimental verification of the theory by pulsed field gradient NMR and tracer zero-length column experiments is discussed.