An attempt is made to interpret NMR data on 3He films for coverages just over one monolayer in terms of motion due to the quantum exchange of particles between layers. A summary of the relevant data and of various possible relaxation mechanisms is given and it is found that a portion of the data seems amenable to an interlayer exchange interpretation. The detailed theory of this process requires the use of the exchange operator concept and a Kubo-theory treatment of the effect of second-layer motion on the exchange process. It is shown that the ldquobarerdquo interlayer exchange process characterized by constant J 12 is slowed by second-layer translational motion so that the effective exchange parameter becomes \~J 12 ap J 12 2/epsiv2, where epsiv2 is a second-layer single-particle translational energy. In order to fit the NMR data it is found that epsiv2 must be evaluated in the classical limit rather than the degenerate Fermi limit, and that \~J 12 ap J 11, the exchange energy within the first-layer solid. These conditions require a helium second-layer effective mass of m * > 5m and J 12>600J 11, which are anomously large values for these