This paper deals with the problem of finding the optimum method of detecting a sine waave of known frequency and amplitude in the presence of noise. The type of noise considered is tshe so-called stationary Gaussian process, which is obtained when thermal noise is passed through an arbitrary linear passive device. The analysis takes into account the fact that in practice only a finite sample of observed signal is available. It is shown that when the noise has a flat spectrum, all the relevant information is contained in the amplitude and phase of the Fourier tansform of the recieved sample at the frequency of the sine wave. Almost the same result holds in the case where the noise has an exponentially decaying auto-correlation, except that in this case the value of teh observed sample at the endpoints of teh sample also play a role.