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.