We present a new measurement of the Hubble Constant H-0 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 HE 0435-1223 from 13-yr light curves obtained as part of the COSMOGRAIL project. Companion papers detail the modelling 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 HE 0435-1223 with previous measurements from systems B1608+656 and RXJ1131-1231 to create a Time Delay Strong Lensing probe (IDSL). In flat A cold dark matter (ACDM) with free matter and energy density, we find H-0 = 71.9(-3.0)(+2.4) km s(-1) Mpc(-1) and Omega(Lambda) = 0.62(-0.35)(+0.24) This measurement is completely independent of, and in agreement with, the local distance ladder measurements of H-0. We explore more general cosmological models combining TDSL with other probes, illustrating its power to break degeneracies inherent to other methods. The joint constraints from IDSL and Planck are H-0 = 69.2(-2.2)(+1.4) km s(-1) Mpc(-1), Omega(Lambda) = 0.70(-0.01)(+0.01) and Omega(k) = 0.003(-0.006)(+0.004) in open ACDM and H-0 = 79.0(-4.2)(+4.4) km s(-1) Mpc(-1), Omega(de) = 0.77(-0.03)(+0.02) and w = -1.38(-0.16)(+0.14) in flat wCDM. In combination 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) in N-eff Lambda CDM and when relaxing the total mass of neutrinos we find Sigma rn(nu) <= 0.182 eV in m(nu) Lambda CDM. Finally, in an open wCDM in combination with Planck and cosmic microwave background lensing, we find H-0 = 77.9(-4.2)(+5.0) km s(-1) Mpc(-1), Omega(de) = 0.77(-0.03)(+0.03), Omega(k) = -0.003(-0.004)(+0.004) and w = -1.37(-0.23)(+0.18).