The authors examine one-dimensional compressible isentropic Navier-Stokes equations with a general pressure law and density-dependent viscosity coefficient when the density connects to vacuum continuously. They prove the global existence and uniqueness of weak solution, and give a decay result for pressure as goes to infinity. It is also proved that no vacuum states and no concentration states develop, and the free boundary do not expand to infinity. The proof is based on the construction of approximate solutions by mollifying initial data and changing the boundary conditions. Then a general compactness argument allows to establish the convergence of approximate solutions.