  The effective action for chiral $W_3$ gravity is studied. It is shown that
the computation of the effective action can be reduced to that of a $SL(3,\re)$
Wess-Zumino-Witten theory. If one assumes that the effective action for the
Wess-Zumino-Witten model is identical to the WZW action up to multiplicative
renormalizations, then the effective action for $W_3$ gravity is, to all
orders, given by a constrained WZW model. The multiplicative renormalization
constants of the WZW model are discussed and it is analyzed which particular
values of these constants are consistent with previous one-loop calculations,
and which reproduce the KPZ formulas for gravity and their generalizations for
$W_3$ gravity.
