A number of distance measures have recently been proposed for the purpose of determining evolutionary similarity among genomes of different species. For each of these measures, a natural but often difficult problem is to determine the \emph{diameter} of the space it defines: What is the maximum distance between any pair of genomes? In this work we study the \emph{syntenic distance} between genomes, introduced by Ferretti, Nadeau, and Sankoff as a way to approximate evolutionary distance between species for which the gene order within chromosomes is not necessarily known. We show that the diameter of the space of $n$-chromosome genomes, with respect to the syntenic distance, is exactly $2n - 4$. The proof of this result is based on a surprising connection between genome rearrangements and the study of \emph{gossip problems} in communication networks.