A lipid-bilayer membrane in the fluid phase when agitated by ambient fluids not only undergoes a shape fluctuation but also a two-dimensional disordered flow. Thus, besides moving individually, lipid molecules in a two-dimensional fluid element can move collectively in the lateral direction. This collective motion, which has been often overlooked in studying diffusion in a fluid membrane, can shift a lipid molecule as well as its Brownian motion. Calculating the self-diffusion constant of a passive scalar-a particle convected neither affecting the flow nor making the Brownian motion-in a fluid membrane fluctuating around a plane in aqueous environments, we show that the collective motion cannot be neglected in explaining observed values of the self-diffusion constant of a lipid molecule.