We present a new measurement of the Hubble Constant H0 and other cosmological parameters based on the joint analysis of three multiply-imaged quasar systems with measured gravitational time delays. First, we measure the time delay of HE0435-1223 from 13-year light curves obtained as part of the COSMOGRAIL project. Companion papers detail the modeling of the main deflectors and line of sight effects, and how these data are combined to determine the time-delay distance of HE 0435-1223. Crucially, the measurements are carried out blindly with respect to cosmological parameters in order to avoid confirmation bias. We then combine the time-delay distance of HE0435-1223 with previous measurements from systems B1608+656 and RXJ1131-1231 to create a Time Delay Strong Lensing probe (TDSL). In flat $Lambda$CDM with free matter and energy density, we find $H_0$ = 71.9 +2.4 -3.0 km/s/Mpc and $Omega_{Lambda}$ = 0.62 +0.24 -0.35 . This measurement is completely independent of, and in agreement with, the local distance ladder measurements of H0. We explore more general cosmological models combining TDSL with other probes, illustrating its power to break degeneracies inherent to other methods. The TDSL and Planck joint constraints are $H_0$ = 69.2 +1.4 -2.2 km/s/Mpc, $Omega_{Lambda}$ = 0.70 +0.01 -0.01 and $Omega_k$ = 0.003 +0.004 -0.006 in open $Lambda$CDM and $H_0$ = 79.0 +4.4 -4.2 km/s/Mpc, $Omega_{de}$ = 0.77 +0.02 -0.03 and $w$ = -1.38 +0.14 -0.16 in flat $w$CDM. Combined with Planck and Baryon Acoustic Oscillation data, when relaxing the constraints on the numbers of relativistic species we find $N_{eff}$ = 3.34 +0.21 -0.21 and when relaxing the total mass of neutrinos we find 0.182 eV. In an open $w$CDM in combination with Planck and CMB lensing we find $H_0$ = 77.9 +5.0 -4.2 km/s/Mpc, $Omega_{de}$ = 0.77 +0.03 -0.03, $Omega_k$ = -0.003 +0.004 -0.004 and $w$ = -1.37 +0.18 -0.23.
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