We consider a model of the reaction $X+Y\to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent continuous time, simple symmetric random walks. $Y$ particles are transformed instantaneously to $X$ particles upon contact. We start with a fixed number $a\ge 1$ of $Y$ particles at each site to the right of the origin, and define a class of configurations of the $X$ particles to the left of the origin having a finite $l^1$ norm with a specified exponential weight. Starting from any configuration of $X$ particles to the left of the origin within such a class, we prove a central limit theorem for the position of the rightmost visited site of the $X$ particles.