This article is concerned with a new method for the approximate evaluation of Fourier sine and cosine transforms. We develop and analyse a new quadrature rule for Fourier sine and cosine transforms involving transforming the integral to one over the entire real line and then using the trapezoidal rule in order to approximate the transformed integral. This method follows on from the work of Ooura and Mori. A complete error analysis is made using contour integration. An example is examined in detail and the error is analysed using residues and the saddle point method. The method we have developed is characterised by its simplicity and single exponential convergence.