  We present new measurements of the flux power-spectrum P(k) of the $z<0.5$ HI
Lyman-$\alpha$ forest spanning scales k ~ 0.001-0.1 s/km. These results were
derived from 65 far ultraviolet quasar spectra (resolution R~18000) observed
with the Cosmic Origin Spectrograph (COS) on board the Hubble Space Telescope.
The analysis required careful masking of all contaminating, coincident
absorption from HI 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 HI
photoionization rate ($\Gamma_{\rm HI}$) in the z<0.5 intergalactic medium. Our
results confirm most of the previous $\Gamma_{\rm HI}$ estimates. We conclude
that previous concerns of a photon underproduction crisis are now resolved by
demonstrating that the measured $\Gamma_{\rm HI}$ 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.
