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 COSMOGRAILs 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.
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