  Strong gravitational lenses with measured time delays between the multiple
images allow a direct measurement of the time-delay distance to the lens, and
thus a measure of cosmological parameters, particularly the Hubble constant,
$H_{0}$. We present a blind lens model analysis of the quadruply-imaged quasar
lens HE 0435-1223 using deep Hubble Space Telescope imaging, updated time-delay
measurements from the COSmological MOnitoring of GRAvItational Lenses
(COSMOGRAIL), a measurement of the velocity dispersion of the lens galaxy based
on Keck data, and a characterization of the mass distribution along the line of
sight. HE 0435-1223 is the third lens analyzed as a part of the $H_{0}$ Lenses
in COSMOGRAIL's Wellspring (H0LiCOW) project. We account for various sources of
systematic uncertainty, including the detailed treatment of nearby perturbers,
the parameterization of the galaxy light and mass profile, and the regions used
for lens modeling. We constrain the effective time-delay distance to be
$D_{\Delta t} = 2612_{-191}^{+208}~\mathrm{Mpc}$, a precision of 7.6%. From HE
0435-1223 alone, we infer a Hubble constant of $H_{0} =
73.1_{-6.0}^{+5.7}~\mathrm{km~s^{-1}~Mpc^{-1}}$ assuming a flat $\Lambda$CDM
cosmology. The cosmographic inference based on the three lenses analyzed by
H0LiCOW to date is presented in a companion paper (H0LiCOW Paper V).
