  Let $\Gamma$ be a dense subgroup of a simply connected nilpotent Lie group
$G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for
the word metric induced by $S$ on $\Gamma$. We show that $S_n$ (with uniform
measure) becomes equidistributed on $G$ with respect to the Haar measure as n
tends to infinity. We give rates and also prove the analogous result for random
walk averages (i.e. the local limit theorem).
