A digital shaper for high-count-rate detection and amplitude measurement of pulses is proposed and analysed in this paper. The proposed shaper converts pulses with a short leading edge and a long exponential tail into a true Gaussian form. The width of Gaussian pulses can be several times smaller than the rise time of the input pulses, i.e. considerably shorter than the undistorted output pulses provided by standard shapers. Therefore, the proposed true Gaussian shaper resolves strongly overlapped pulses better and provides a higher output count rate. The capabilities of the proposed true Gaussian shaper are analysed with real and simulated output signals of a silicon drift detector of soft X-ray radiation, operating at a high count rate of the collected quanta. Our analysis shows that true Gaussian shapers can increase the count rate of spectrometer systems several times compared with the widely used trapezoidal shapers, while maintaining their amplitude resolution.