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