  Let $G$ be a discrete group, $\mu$ a measure on $G$ and $X$ a proper CAT(0)
space. We show that if $G$ acts non-elementarily with a rank one element on
$X$, then the pushforward $\{Z_n o \}_n$ to $X$ of the random walk generated by
$\mu$ converges almost surely to a rank one point of the boundary. We also show
that in this context, there is a unique stationary measure on the visual
boundary $\partial_\infty X$ of $X$, and that the drift of the random walk is
almost surely positive.
