Let $U$ be a multiply connected region in $\mathbf{R}^ 2$ with
smooth boundary. Let $P_\epsilon$ be a polyomino in $\epsilon\mathbf{Z}^2$
approximating $U$ as $\epsilon \to 0$.We show that, for certain
boundary conditions on $P_\eqsilon$, the height distribution on a random domino
tiling (dimer covering) of $P_\eqsilon$ is conformally invariant in the limit
as $\epsilon$ tends to 0, in the sense that the distribution of heights of
boundary components (or rather, the difference of the heights from their mean
values) only depends on the conformal type of $U$. The mean height is not
strictly conformally invariant but transforms analytically under conformal
mappings in a simple way. The mean height and all the moments are explicitly
evaluated.
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