We have studied the influence of the local density of optical states (LDOS) on the rate and efficiency of Forster resonance energy transfer (FRET) from a donor to an acceptor. The donors and acceptors are dye molecules that are separated by a short strand of double-stranded DNA. The LDOS is controlled by carefully positioning the FRET pairs near a mirror. We find that the energy transfer efficiency changes with LDOS, and that, in agreement with theory, the energy transfer rate is independent of the LDOS, which allows one to quantitatively control FRET systems in a new way. Our results imply a change in the characteristic Forster distance, in contrast to common lore that this distance is fixed for a given FRET pair.