We propose a fully discrete numerical scheme for solving an advection– diffusion equation on a family of evolving hypersurfaces. The method is based on a diffuse interface approach that combines a level set description of the moving surfaces with a discretization by linear finite elements in space and a backward Euler scheme in time. We estimate the error in L∞(L2) and L2(H1) in terms of the spatial grid size h, the time step τ and the thickness of the diffuse interface. Furthermore, we present the results of test calculations that support our analysis.