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", 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, 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 open the way to
utilize time delays in lens galaxy clusters as an important alternative tool
for measuring the expansion rate and the geometry of the Universe.