  Let $X$ be a proper, geodesically complete Hadamard space, and $\
\Gamma<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed
point of a rank one isometry of $X$ in its infinite limit set. In this paper we
prove that if $\Gamma$ has non-arithmetic length spectrum, then the Ricks'
Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis
measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks'
Bowen-Margulis measure is finite, then we also have equidistribution of
$\Gamma$-orbit points in $X$, which in particular yields an asymptotic estimate
for the orbit counting function of $\Gamma$. This generalizes well-known facts
for non-elementary discrete isometry groups of Hadamard manifolds with pinched
negative curvature and proper CAT$(-1)$-spaces.
