We use a result from the theory of geometric representation of graphs to show that the separator of a fullerene graph on n vertices cannot exceed 24/n, thus improving the best currently known upper bound of 1-3/n. The result is then combined with a recently established upper bound on the smallest eigenvalue of fullerene graphs to show that there are only finitely many Ramanujan fullerenes. That settles down a conjecture on fullerenes made by the Graffiti software.