  In this paper, we are concerned with the Cauchy problem for one-dimensional
compressible isentropic Navier-Stokes equations with density-dependent
viscosity $\mu(\rho)=\rho^\alpha (\alpha>0)$ and pressure
$P(\rho)=\rho^{\gamma}\ (\gamma>1)$. We will establish the global existence and
asymptotic behavior of weak solutions for any $\alpha>0$ and $\gamma>1$ under
the assumption that the density function keeps a constant state at far fields.
This enlarges the ranges of $\alpha$ and $\gamma$ and improves the previous
results presented by Jiu and Xin. As a result, in the case that
$0<\alpha<\frac12$, we obtain the large time behavior of the strong solution
obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).
