From the introduction: We establish the global existence and uniqueness of weak solutions to the Navier-Stokes equations for a one-dimensional isentropic viscous gas with a jump to the vacuum initially when the viscosity depends on the density: \[ \rho_\tau+(\rho u)_\xi=0, \quad(\rho u)_\tau +(\rho u^2+P(\rho))_\xi=\bigl(\mu(\rho) u_\xi)_\xi, \] in \(\tau>0\), \(a(\tau)<\xi< b(\tau)\), where \(\rho\), \(u\) and \(P(\rho)\) are the density, the velocity and the pressure, respectively, \(\mu(\rho)>0\) is the viscosity coefficient, \(a(\tau)\) and \(b(\tau)\) are the free boundaries, i.e., the interface of the gas and the vacuum.