The exclusive production cross sections for and mesons have been measured in proton-proton reactions at p lab ϭ3.67 GeV/c. The observed / cross section ratio is (3.8Ϯ0.2 Ϫ0.9 ϩ1.2 )ϫ10 Ϫ3 . After phase space corrections, this ratio is enhanced by about an order of magnitude relative to naive predictions based upon the Okubo-Zweig-Iizuka ͑OZI͒ rule, in comparison to an enhancement by a factor ϳ3 previously observed at higher beam momenta. The modest increase of this enhancement near the production threshold is compared to the much larger increase of the / ratio observed in specific channels of p p annihilation experiments. Furthermore, differential cross section results are also presented which indicate that although the meson is predominantly produced from a 3 P 1 proton-proton entrance channel, other partial waves contribute significantly to the production mechanism at this beam momentum.