We consider the dimer model on the Aztec diamond with Fock's weights, which is gauge equivalent to the model with any choice of positive weight function. We prove an explicit, compact formula for the inverse Kasteleyn matrix, thus extending numerous results in the case of periodic graphs. We also show an explicit product formula for the partition function; as a specific instance of the genus 0 case, we recover Stanley's formula. We then use our explicit formula for the inverse Kasteleyn matrix to recover, in a simple way, limit shape results; we also obtain new ones. In doing so, we extend the correspondence between the limit shape and the amoeba of the corresponding spectral curve of arXiv:2306.07482 to the case of non-generic weights.