Rigidity percolation (RP) is the emergence of mechanical stability in networks. Motivated by the experimentally observed fractal nature of materials like colloidal gels and disordered fiber networks, we study RP in a fractal network where intrinsic correlations in particle positions is controlled by the fractal iteration. Specifically, we calculate the critical packing fractions of site-diluted lattices of Sierpiński gaskets (SG's) with varying degrees of fractal iteration. Our results suggest that although the correlation length exponent and fractal dimension of the RP of these lattices are identical to that of the regular triangular lattice, the critical volume fraction is dramatically lower due to the fractal nature of the network. Furthermore, we develop a simplified model for an SG lattice based on the fragility analysis of a single SG. This simplified model provides an upper bound for the critical packing fractions of the full fractal lattice, and this upper bound is strictly obeyed by the disorder averaged RP threshold of the fractal lattices. Our results characterize rigidity in ultralow-density fractal networks.