This is a continuation of the article (Comm. Partial Differential Equations 26 (2001) 965). In this article, the authors consider the one-dimensional compressible isentropic Navier--Stokes equations with gravitational force, fixed boundary condition, a general pressure and the density-dependent viscosity coefficient when the viscous gas connects to vacuum state with a
jump in density. Precisely, the viscosity coefficient μ is proportional to ρ^θ and 0 < θ < 1/2, where ρ is the density, and the pressure P=P(ρ) is a general pressure. The global existence and the uniqueness of weak solution are proved.