We observe directly the diffusion and aggregation of buckyball clusters dispersed in thin, freely-suspended films of smectic liquid crystal using reflected light microscopy. The buckyballs at early times are barely resolvable, nanoscale clusters, which diffuse stochastically in the film. Clusters eventually coalesce to form micron-scale, fractal aggregates whose effective radius increases approximately linearly with time, so that after several days, millimeter-size fractals extend across the entire film. The measured fractal dimension of these final networks suggests that the aggregation of the buckyball clusters in the film is a diffusion-limited process.