  Let $\mf g$ be the Witt algebra or the positive Witt algebra. It is well
known that the enveloping algebra $U(\mf g )$ has intermediate growth and thus
infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of
$U(\mf g)$ is {\em just infinite} in the sense that any proper quotient of
$U(\mf g)$ has polynomial growth.
  This proves a conjecture of Petukhov and the second named author for the
positive Witt algebra.
  We also establish the corresponding results for quotients of the symmetric
algebra $S(\mf g)$ by proper Poisson ideals.
  In fact, we prove more generally that any central quotient of the universal
enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We
give several applications. In particular, we easily compute the annihilators of
Verma modules over the Virasoro algebra.
