  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.
