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We present new measurements of the flux power-spectrum P (k) of the z < 0.5 H   i Lyman-α Forest spanning scales |$k \sim 0.001\!-\!0.1\, \mathrm{s \, km}^{-1}$|⁠. These results were derived from 65 far-ultraviolet quasar spectra (resolution |$R \sim 18\, 000$|⁠) observed with the Cosmic Origin Spectrograph (COS) on board the Hubble Space Telescope. The analysis required careful masking of all contaminating, coincident absorption from H   i and metal–line transitions of the Galactic interstellar medium and intervening absorbers as well as proper treatment of the complex COS line-spread function. From the P (k) measurements, we estimate the H   i photoionization rate (⁠|$\Gamma _{\rm H\,{\small I}}$|⁠) in the z < 0.5 intergalactic medium. Our results confirm most of the previous |$\Gamma _{\rm H\,{\small I}}$| estimates. We conclude that previous concerns of a photon underproduction crisis are now resolved by demonstrating that the measured |$\Gamma _{\rm H\,{\small I}}$| can be accounted for by ultraviolet emission from quasars alone. In a companion paper, we will present constraints on the thermal state of the z < 0.5 intergalactic medium from the P (k) measurements presented here. [ABSTRACT FROM AUTHOR]