  This article is concerned with the analysis of the one-dimensional
compressible Euler equations with a singular pressure law, the so-called hard
sphere equation of state. The result is twofold. First, we establish the
existence of bounded weak solutions by means of a viscous regularization and
refined compensated compactness arguments. Second, we investigate the smooth
setting by providing a detailed description of the impact of the singular
pressure on the breakdown of the solutions. In this smooth framework, we
rigorously justify the singular limit towards the free-congested Euler
equations, where the compressible (free) dynamics is coupled with the
incompressible one in the constrained (i.e. congested) domain.
