We present a 63 ks FUSE observation of the narrow-line Seyfert 1 galaxy Arakelian 564. The spectrum is dominated by the strong emission in the O VI lambdalambda1032, 1038 resonance doublet. Strong, heavily saturated absorption troughs due to Lyman series of hydrogen, O VI and C III lambda977 at velocities near the systemic redshift of Ark 564 are also observed. We used the column densities of O VI and C III in conjunction with the published column densities of species observed in the UV and X-ray bands, to derive constraints on the physical parameters of the absorber through photoionization modeling. The available data suggest that the UV and X-ray absorbers in Ark 564 are physically related and possibly identical. The combination of constraints indicates that the absorber is characterized by a narrow range in total column density N-H and U, centered at log N-H approximate to 21 and log U approximate to -1.5, and may be spatially extended along the line of sight.