  Strong-lensing time delays enable measurement of the Hubble constant
($H_{0}$) 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 $H_0$
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
$H_0$, effectively implementing a maximally flexible mass model. Following a
blind analysis, we estimated the angular diameter distance to the lens galaxy
$D_{\rm d} = 865_{-81}^{+85}$ Mpc and the time-delay distance $D_{\Delta t} =
2180_{-271}^{+472}$ Mpc, giving $H_0 = 77.1_{-7.1}^{+7.3}$ km s$^{-1}$
Mpc$^{-1}$ - 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 the 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 ($H_0 = 78.3^{+3.4}_{-3.3}$ km s$^{-1}$
Mpc$^{-1}$) and for seven lenses ($H_0 = 74.2_{-1.6}^{+1.6}$ km s$^{-1}$
Mpc$^{-1}$), or for seven lenses using single-aperture kinematics and the same
maximally flexible models used by us ($H_0 = 73.3^{+5.8}_{-5.8}$ km s$^{-1}$
Mpc$^{-1}$). This agreement corroborates the methodology of time-delay
cosmography.
