We are often faced with the task of having to estimate the amplitude of a source signal in the presence of a background. In the simplest case, the background can be taken as being flat, and of unknown magnitude B, and the source signal of interest assumed to be the amplitude A of a peak of known shape and position. We present a robust method to find the most probable values of A and B by applying the one-dimensional Newton-Raphson method. In the derivation of the formula, we adopted the Bayesian statistics and assmumed Poisson distribution so that the results could be applied to the analysis of very weak signals, as observed in FIMS (Far-ultraviolet IMaging Spectrogaph).