Solitons are studied in a model of a fiber Bragg grating (BG) whose local reflectivity is subjected to periodic modulation. The superlattice opens an infinite number of new bandgaps in the model's spectrum. Averaging and numerical continuation methods show that each gap gives rise to gap solitons (GSs), including asymmetric and double-humped ones, which are not present without the superlattice. Computation of stability eigenvalues and direct simulations reveal the existence of completely stable families of fundamental GSs filling the new gaps also at negative frequencies, where the ordinary GSs are unstable. Moving stable GSs with positive and negative effective mass are found too.