Strong-lensing time delays enable the measurement of the Hubble constant (H0) independently of other traditional methods. The main limitation to the precision of time-delay cosmography is mass-sheet degeneracy (MSD). Some of the previous TDCOSMO analyses broke the MSD by making standard assumptions about the mass density profile of the lens galaxy, reaching 2% precision from seven lenses. However, this approach could potentially bias the H0 measurement or underestimate the errors. For this work, we broke the MSD for the first time using spatially resolved kinematics of the lens galaxy in RXJ1131-1231 obtained from the Keck Cosmic Web Imager spectroscopy, in combination with previously published time delay and lens models derived from Hubble Space Telescope imaging. This approach allowed us to robustly estimate H0, effectively implementing a maximally flexible mass model. Following a blind analysis, we estimated the angular diameter distance to the lens galaxy D_d_=865_-81_^+85^Mpc and the time-delay distance D_{DELTA}t_=2180_-271_^+472^Mpc, giving H0=77.1_-7.1_^+7.3^km/s/Mpc for a flat {LAMBDA} cold dark matter cosmology. The error budget accounts for all uncertainties, including the MSD inherent to the lens mass profile and line-of-sight effects, and those related to the mass-anisotropy degeneracy and projection effects. Our new measurement is in excellent agreement with those obtained in the past using standard simply parametrized mass profiles for this single system (H0=78.3_-3.3_^+3.4^km/s/Mpc) and for seven lenses (H0=74.2_-1.6_^+1.6^km/s/Mpc), or for seven lenses using single-aperture kinematics and the same maximally flexible models used by us (H0=73.3_-5.8_^+5.8^km/s/Mpc). This agreement corroborates the methodology of time-delay cosmography.