We present a proof of a conjecture about the relationship between Baxter permutations and pairs of alternating sign matrices that are produced from domino tilings of Aztec diamonds. It is shown that a tiling corresponds to a pair of ASMs that are both permutation matrices if and only if the larger permutation matrix corresponds to a Baxter permutation. There has been a thriving literature on both pattern-avoiding permutations of various kinds [Baxter 1964, Dulucq and Guibert 1988] and tilings of regions using dominos or rhombuses as tiles [Elkies et al. 1992, Kuo 2004]. However, there have not as of yet been many links between these two areas of enumerative combinatorics. This paper gives one such link...