  We analyze the total cross section data for $pp \to pp\omega$ near threshold
measured recently at SATURNE. Using an effective range approximation for the
on-shell $pp$ S-wave final state interaction we extract from these data the
modulus $|\Omega| = 0.53$ fm$^4$ of the threshold transition amplitude
$\Omega$. We present a calculation of various (tree-level) meson exchange
diagrams contributing to $\Omega$. It is essential that $\omega$-emission from
the anomalous $\omega\rho\pi$-vertex interferes destructively with
$\omega$-emission from the proton lines. The contribution of scalar
$\sigma$-meson exchange to $\Omega$ turns out to be negligibly small. Without
introducing off-shell meson-nucleon form factors the experimental value
$|\Omega|=0.53$ fm$^4$ can be reproduced with an $\omega N$-coupling constant
of $g_{\omega N}=10.7$. The results of the present approach agree qualitatively
with the J\"ulich model. We also perform a combined analysis of the reactions
$pp\to pp\pi^0, pn\pi^+, pp\eta, pp\omega$ and $pn\to pn\eta$ near threshold.
