Unlike graphene and similar structures, 2D covalent organic frameworks (2D COFs) exhibit intrinsic porosity with a high areal density of well-defined and uniform openings. Given the pore size adjustability, 2D COFs are likely to outperform artificially perforated inorganic layers with respect to their prospects in membrane separation. Yet, exploring the mass transport in 2D COFs is hidden by the lack of laterally extended free-standing membranes. This work reports on direct molecular permeation measurements with single crystals of an interfacially synthesized boronate ester 2D COF. In accordance with the material topography, the atmospheric and noble gases readily pass the suspended nanosheets while their areal porosity is quantified to be almost 40% exceeding that in any 2D membranes known. However, bulkier aromatic hydrocarbons are found to deviate substantially from Graham's law of diffusion. Counterintuitively, the permeation rate is demonstrated to rise from benzene to toluene and further to xylene despite the increase in the molecular mass and dimensions. The results are interpreted in terms of adsorption-mediated flow that appears to be an important transport mechanism for microporous planar nanomaterials.