  We give a complete account of the Schr\"odinger representation approach to
the calculation of the Weyl anomaly of ${\cal N}=4$ SYM from the AdS/CFT
correspondence. On the AdS side, the $1/N^2$ correction to the leading order
result receives contributions from all the fields of Type IIB Supergravity, the
contribution of each field being given by a universal formula. The correct
matching with the CFT result is thus a highly non-trivial test of the
correspondence.
