The frequency-domain exponential transfer function of a delay function cannot be realized with a finite number of lumped elements. Therefore an approximation of a rational quotient of polynomials has to be used. While the use of Bessel polynomials results in the well-known all-pole Bessel-Thomson approximation, a Taylor expansion of the exponential transfer function of a delay around one point results in another type of rational transfer, known as Pade approximation. Although a Bessel-Thomson approximation results in an overshoot-free step response it has slower response and smaller bandwidth in comparison to a Pade-approximated delay. Unfortunately, the latter suffers from overshoot. To reduce the overshoot but preserve the fast-response and large-bandwidth properties, a new delay approximation method is introduced. The method is based on approximation of the delta time-domain response of an ideal delay by a narrow Gaussian time-domain impulse response. The subsequent Pade approximation of the corresponding Gaussian transfer function yields a rational transfer function that is ready for implementation in an analog fashion and realizes a delay with both a large bandwidth and little overshoot.