The asymptotic behavior of the local time of Brownian motion on the circle is investigated. For fixed time point $t$ this is a (random) continuous function on $S^1$. It is shown that after appropriate norming the distribution of this random element in $C(S^1)$ converges weakly as $t \rightarrow \infty$. The limit is identified as $2(B(x) - \int B(y) dy)$ where $B$ is the Brownian bridge. The result is applied to obtain the asymptotic distribution of a Cramer-von Mises type statistic for the global deviation of the local time from the constant $t$ on $S^1$.