Fast and high-quality Gaussian random number generation (GRNG) is a key capability for simulations across a wide range of disciplines. In this article, we present an enhanced Box-Muller method for GRNG using one uniform variable. Its probability density function (PDF) is given in closed form as a function of one parameter. In this article, the theoretical basis of this method is quite thoroughly discussed and is evaluated using several different statistical tests, including the chi-square test and the Anderson-Darling test. The simulations results show good performances of this method which generates accurately a true Gaussian PDF even at very high σ (standard deviations) values in comparison with the standard Box-Muller method.