This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity. In particular, we focus our attention on the viscosity taking the form μ(ρ) = ρϵ(ϵ > 0). For the selected density-dependent viscosity, it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates, in addition to the scaling argument and elementary energy analysis. Moreover, our results extend the studies in [Xin Z P. Comm Pure Appl Math, 1993, 46(5): 621–665].