  The scalar field exchange diagram for the correlation function of four scalar
operators is evaluated in anti-de Sitter space, $AdS_{d+1}$. The conformal
dimensions $\Delta_i$, $i=1,...,4$ of the scalar operators and the dimension
$\Delta$ of the exchanged field are arbitrary, constrained only to obey the
unitarity bound. Techniques similar to those developed earlier for gauge boson
exchange are used, but results are generally more complicated. However, for
integer $\Delta_i, \Delta$, the amplitude can be presented as a multiple
derivative of a simple universal function. Results simplify if further
conditions hold, such as the inequalities, $\Delta< \Delta_1+\Delta_3$ or
$\Delta<\Delta_2+\Delta_4$. These conditions are satisfied, with $<$ replaced
by $\le$, in Type IIB supergravity on $AdS_5\times S_5$ because of selection
rules from SO(6) symmetry. A new form of interaction is suggested for the
marginal case of the inequalities. The short distance asymptotics of the
amplitudes are studied. In the direct channel the leading singular term agrees
with the double operator product expansion. Logarithmic singularities occur at
sub-leading order in the direct channel but at leading order in the crossed
channel. When the inequalities above are violated, there are also $(\log)^2$
singularities in the direct channel.
