Dynamic Monte Carlo simulation of a bead-spring model of flexible macromolecules threading through a very narrow pore in a very thin rigid membrane are presented, assuming at the cis side of the membrane a purely repulsive monomer-wall interaction, while the trans side is attractive. Two choices of monomer-wall attraction epsilon are considered, one choice is slightly below and the other slightly above the "mushroom to pancake" adsorption threshold epsilon(c) for an infinitely long chain. Studying chain lengths N=32, 64, 128, and 256 and varying the number of monomers N-trans (time t=0) that have already passed the pore when the simulation started, over a wide range, we find for epsilonepsilon(c) a finite number N-trans(t=0) suffices that the translocation probability is close to unity. In the case epsilonepsilon(c), we find that the translocation time scales as tauproportional toN(1.65+/-0.08). We suggest a tentative scaling explanation for this result. Also the distribution of translocation times is obtained and discussed. (C) 2004 American Institute of Physics.