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 analysed as a part of the H-0 Lenses in COSMOGRAIL's Wellspring (HOLiCOW) project. We account for various sources of systematic uncertainty, including the detailed treatment of nearby perturbers, the parametrization of the galaxy light and mass profile, and the regions used for lens modelling. We constrain the effective time delay distance to be D-Delta t = 2612(191)(+208) Mpc, a precision of 7.6 per cent. From HE 0435-1223 alone, we infer a Hubble constant of H-0 = 73.1(6.0)(+5.7) km s(-1) Mpc(-1) assuming a flat ACDM cosmology. The cosmographic inference based on the three lenses analysed by HOLiCOW to date is presented in a companion paper (HOLiCOW Paper V).