This article presents a Bayesian algorithm for detection and tracking of a target using the track-before-detect framework. This strategy enables to detect weak targets and to circumvent the data association problem originating from the detection stage of classical radar systems. We first establish a Bayesian recursion, which propagates the target state probability density function. Since raw measurements are generally related to the target state through a nonlinear observation function, this recursion does not admit a closed form expression. Therefore, in order to obtain a tractable formulation, we propose a Gaussian mixture approximation. Our targeted application is passive radar, with civilian broadcasters used as illuminators of opportunity. Numerical simulations show the ability of the proposed algorithm to detect and track a target at very low signal-to-noise ratios.