Molecular dynamics computer simulations of the vapor/liquid phase transition of a Lennard-Jones fluid are presented. The simulations reveal that, in the late stage of the transition, clusters grow linearly with time. This linear growth regime can be described by an extension of the Lifshitz-Slyozov theory with an asymptotic droplet growth power law t(1/3). This growth mode is interrupted by the coalescence of clusters, whose motions have Brownian character. The Brownian nature of the cluster trajectories is a consequence of internal degrees of freedom rather than of environmental noise. The diffusion coefficients of these clusters, as a function of cluster size, are found to be well represented by the kinetic theory of the ideal gas.