We demonstrate a method that allows a high-efficiency single-photon-avalanche diode (SPAD) with a thick absorption region (>10 mu m) to count single photons at rates significantly higher than previously demonstrated. We apply large (>30V) AC bias gates to the SPAD at 1GHz and detect minute avalanches with a discrimination threshold of 5(1) mV by means of radio frequency interferometry. We measure a reduction by a factor of approximate to 500 in the average charge per avalanche when compared to operation in its traditional active-quenching module and a relative increase in >19\% in detection efficiency at 850nm. The reduction in charge strongly suppresses self-heating effects in the diode that can degrade performance at high avalanche rates. We show that the single-photon detection system maintains high efficiency at count rates exceeding 10(8) s(-1).