  We compute completely the BRST--antibracket cohomology on local functionals
in two-dimensional Weyl invariant gravity for given classical field content
(two dimensional metric and scalar matter fields) and gauge symmetries (two
dimensional diffeomorphisms and local Weyl transformations). This covers the
determination of all classical actions, of all their rigid symmetries, of all
background charges and of all candidate gauge anomalies. In particular we show
that the antifield dependence can be entirely removed from the anomalies and
that, if the target space has isometries, the condition for the absence of
matter field dependent Weyl anomalies is more general than the familiar
`dilaton equations'.
