In this letter, we report the first resonant-cavity-enhanced single-photon avalanche diode (RCE SPAD) fabricated on a reflecting silicon-on-insulator (SOI) substrate. The substrate incorporates a two-period distributed Bragg reflector fabricated using a commercially available double-SOI process. The RCE SPAD detectors have peak photon detection efficiencies ranging from 42% at 780 nm to 34% at 850 nm and time resolution of 35-ps full-width at half-maximum. Typical dark count rates of 450, 3500, and 100 000 c/s were measured at room temperature with RCE SPADs having, respectively 8-, 20-, and 50-mum diameter.