By the Lindeberg principle, we develop in this paper an approximation to one dimensional (possibly) asymmetric α-stable distributions with α∈(0,2) in the smooth Wasserstein distance. It is the first time that the general stable central limit theorem is proved by the Lindeberg principle, and that this theorem with α∈(0,1] is proved by a new method other than Fourier analysis. Our main tools are a Taylor-like expansion and a Kolmogorov forward equation.