By quasistationary approximation for the one-dimensional Stefan problem is meant a pair $T_{{\rm qs}}(x, t)$ (the temperature), $X_{{\rm qs}}(t)$ (the interface location) such that $T_{{\rm qs}}$, instead of satisfying the heat equation $c\rho T_t=kT_{xx}$, is taken to be a linear function of $x$. The condition $X(0)=0$ and the conditions specified on the fixed boundary and on the free boundary are then enough to determine both $T_{{\rm qs}}$ and $X_{{\rm qs}}$. This paper is devoted to proving that the solution $(T, X)$ of the original problem tends to $(T_{{\rm qs}}, X_{{\rm qs}})$ as $c$ tends to zero. \par Some numerical examples are presented and discussed.