The extraordinary absorption cross section and high photoluminescence (PL) quantum yield of perovskite nanocrystals make this type of material attractive to a variety of applications in optoelectronics. For the same reasons, nanocrystals are also ideally suited to function as nanoantennae to excite nearby single dye molecules by fluorescence resonance energy transfer (FRET). Here, we demonstrate that FAPbBr(3) perovskite nanocrystals, of cuboidal shape and approximately 10 nm in size, are capable of selectively exciting single cyanine 3 molecules at a concentration 100-fold higher than standard single-molecule concentrations. This FRET antenna mechanism increases the effective brightness of the single dye molecules 100-fold. Photon statistics and emission polarization measurements provide evidence for the FRET process by revealing photon antibunching with unprecedented fidelity and highly polarized emission stemming from single dye molecules. Remarkably, the quality of single-photon emission improves 1.5-fold compared to emission collected directly from the nanocrystals because the higher excited states of the dye molecule act as effective filters to multiexcitons. The same process gives rise to efficient deshelving of the molecular triplet state by reverse intersystem crossing (RISC), translating into a reduction of the PL saturation of the dye, thereby increasing the maximum achievable PL intensity of the dye by a factor of 3.