  Consider a one-dimensional stepping stone model with colonies of size $M$ and
per-generation migration probability $\nu$, or a voter model on $\mathbb{Z}$ in
which interactions occur over a distance of order $K$. Sample one individual at
the origin and one at $L$. We show that if $M\nu/L$ and $L/K^2$ converge to
positive finite limits, then the genealogy of the sample converges to a pair of
Brownian motions that coalesce after the local time of their difference exceeds
an independent exponentially distributed random variable. The computation of
the distribution of the coalescence time leads to a one-dimensional parabolic
differential equation with an interesting boundary condition at 0.
