A Gaussian statistic, or discriminator, is proposed for conclusively discriminating between two pure quantum states. The discriminator is based on registrations from a limit of an increasing number of weak measurements on a single system. The average error of the discriminator is near that of the optimal conclusive Helstrom discriminator and, in fact, the Gaussian discriminator is the limit point of a class of conclusive discriminators which includes the Helstrom discriminator. The Gaussian discriminator always leaves some post-measurement state (PMS) separation; by contrast, the PMS separation with the Helstrom discriminator is always zero. Simple, closed-form expressions are given for the distribution and error performance of the Gaussian discriminator.