  Let $K$ be a field and let $A$ be a finitely generated prime $K$-algebra. We
generalize a result of Smith and Zhang, showing that if $A$ is not PI and does
not have a locally nilpotent ideal, then the extended centre of $A$ has
transcendence degree at most ${\rm GKdim}(A)-2$ over $K$. As a consequence, we
are able to show that if $A$ is a prime $K$-algebra of quadratic growth, then
either the extended centre is a finite extension of K or $A$ is PI. Finally, we
give an example of a finitely generated non-PI prime $K$-algebra of GK
dimension 2 with a locally nilpotent ideal such that the extended centre has
infinite transcendence degree over $K$.
