This paper introduces a novel recursive Bayesian estimator for the center and radius of a circle based on noisy points. Each given point is assumed to be a noisy measurement of an unknown true point on the circle that is corrupted with known isotropic Gaussian noise. In contrast to existing approaches, the novel method does not make assumptions about the true points on the circle, where the measurements stem from. Closed-form expressions for the measurement update step are derived. Simulations show that the novel method outperforms standard Bayesian approaches for circle fitting.