We report a computer-simulation study of homogeneous gas-liquid nucleation in a Lennard-Jones system. Using umbrella sampling, we compute the free energy of a cluster as a function of its size. A thermodynamic integration scheme is employed to determine the height of the nucleation barrier as a function of supersaturation. Our simulations illustrate that the mechanical and the thermodynamical surfaces of tension and surface tension differ significantly. In particular, we show that the mechanical definition of the surface tension cannot be used to compute this barrier height. We find that the relations recently proposed by McGraw and Laaksonen [J. Chem. Phys. 106, 5284 (1997)] for the height of the barrier and for the size of the critical nucleus are obeyed.