  Soliton gases represent large random soliton ensembles in physical systems
that display integrable dynamics at the leading order. Despite significant
theoretical developments and observational evidence of ubiquity of soliton
gases in fluids and optical media their controlled experimental realization has
been missing. We report the first controlled synthesis of a dense soliton gas
in deep-water surface gravity waves using the tools of nonlinear spectral
theory (inverse scattering transform (IST)) for the one-dional focusing
nonlinear Schr\"odinger equation. The soliton gas is experimentally generated
in a one-dimensional water tank where we demonstrate that we can control and
measure the density of states, i. e. the probability density function
parametrizing the soliton gas in the IST spectral phase space. Nonlinear
spectral analysis of the generated hydrodynamic soliton gas reveals that the
density of states slowly changes under the influence of perturbative
higher-order effects that break the integrability of the wave dynamics.
