Transport by extremely weak turbulence occurring on nights with clear skies and weak winds is examined from seven tower levels of eddy-correlation data taken from each of two field programs. The very small flux is systematic, provided that the perturbations are computed from a record-dependent averaging length, which must be as small as 10 s in very stable conditions. With traditional methods for computing the flux, these fluxes were considered too small to estimate, in that the computed values behaved erratically. For extremely weak turbulence, the fluxes decrease systematically with height and often indicate very shallow boundary-layer depths on the order of 10 m. However, in one field program, the turbulence slowly increases with height above the surface flux-based boundary layer apparently due to horizontal advection of stronger turbulence driven by modest surface heterogeneity. For very weak turbulence, the eddy diffusivity for momentum is systematically greater than that for heat in both field programs. The dependence of the turbulence strength and its variability with stability is examined in some detail.