  We present a blind time-delay cosmographic analysis for the lens system DES
J0408$-$5354. This system is extraordinary for the presence of two sets of
multiple images at different redshifts, which provide the opportunity to obtain
more information at the cost of increased modelling complexity with respect to
previously analyzed systems. We perform detailed modelling of the mass
distribution for this lens system using three band Hubble Space Telescope
imaging. We combine the measured time delays, line-of-sight central velocity
dispersion of the deflector, and statistically constrained external convergence
with our lens models to estimate two cosmological distances. We measure the
"effective" time-delay distance corresponding to the redshifts of the deflector
and the lensed quasar $D_{\Delta t}^{\rm eff}=3382^{+146}_{-115}$ Mpc and the
angular diameter distance to the deflector $D_{\rm d}=1711^{+376}_{-280}$ Mpc,
with covariance between the two distances. From these constraints on the
cosmological distances, we infer the Hubble constant $H_0 = 74.2^{+2.7}_{-3.0}$
km s$^{-1}$ Mpc$^{-1}$ assuming a flat $\Lambda$CDM cosmology and a uniform
prior for $\Omega_{\rm m}$ as $\Omega_{\rm m} \sim \mathcal{U}(0.05, 0.5)$.
This measurement gives the most precise constraint on $H_0$ to date from a
single lens. Our measurement is consistent with that obtained from the previous
sample of six lenses analyzed by the $H_0$ Lenses in COSMOGRAIL's Wellspring
(H0LiCOW) collaboration. It is also consistent with measurements of $H_0$ based
on the local distance ladder, reinforcing the tension with the inference from
early Universe probes, for example, with 2.2$\sigma$ discrepancy from the
cosmic microwave background measurement.
