  We use power-counting arguments as an organizing principle to apply chiral
perturbation theory, including an explicit $\Delta$, to the $p p \rightarrow p
p \pi^0$ reaction near threshold. There are two lowest-order leading mechanisms
expected to contribute to the amplitude with similar magnitudes: an impulse
term, and a $\Delta$-excitation mechanism. We examine formally sub-leading but
potentially large mechanisms, including pion-rescattering and short-ranged
contributions. We show that the pion-rescattering contribution is enhanced by
off-shell effects and has a sign opposite to that of a recent estimate based on
a PCAC pion interpolating field. Our result is that the impulse term interferes
destructively with the pion rescattering and $\Delta$-excitation terms. In
addition, we have modeled the short-ranged interaction using $\sigma$ and
$\omega$ exchange mechanisms. A recoil correction to the impulse approximation
is small. The total amplitude obtained including all of these processes is
found to yield cross sections substantially smaller than the measured ones.
