  Realizing Refsdal's original idea from 1964, we present estimates of the
Hubble constant that are complementary to and potentially competitive with
those of other cosmological probes. We use the observed positions of 89
multiple images, with extensive spectroscopic information, from 28 background
sources and the measured time delays between the images S1-S4 and SX of
supernova "Refsdal" ($z = 1.489$), which were obtained thanks to Hubble Space
Telescope (HST) deep imaging and Multi Unit Spectroscopic Explorer (MUSE) data.
We extend the strong lensing modeling of the Hubble Frontier Fields (HFF)
galaxy cluster MACS J1149.5$+$2223 ($z = 0.542$), published by Grillo et al.
(2016), and explore different $\Lambda$CDM models. Taking advantage of the
lensing information associated to the presence of very close pairs of multiple
images at various redshifts and to the extended surface brightness distribution
of the SN Refsdal host, we can reconstruct the total mass density profile of
the cluster very precisely. The combined dependence of the multiple image
positions and time delays on the cosmological parameters allows us to infer the
values of $H_{0}$ and $\Omega_{\rm m}$ with relative (1$\sigma$) statistical
errors of, respectively, 6% (7%) and 31% (26%) in flat (general) cosmological
models, assuming a conservative 3% uncertainty on the final time delay of image
SX and, remarkably, no priors from other cosmological experiments. Our best
estimate of $H_{0}$, based on the model described in this work, will be
presented when the final time-delay measurement becomes available. Our results
show that it is possible to utilize time delays in lens galaxy clusters as an
important alternative tool for measuring the expansion rate and the geometry of
the Universe.
