Belief propagation (BP) is a well-celebrated iterative optimization algorithm in statistical learning over network graphs with vast applications in many scientific and engineering fields. This paper studies a fundamental property of this algorithm, namely, its convergence behaviour. Our study is conducted through the problem of distributed state estimation for a networked linear system with additive Gaussian noises, using the weighted least-squares criterion. The corresponding BP algorithm is known as Gaussian BP. Our main contribution is to show that Gaussian BP is guaranteed to converge, under a mild regularity condition. Our result significantly generalizes previous known results on BP's convergence properties, as our study allows general network graphs with cycles and network nodes with random vectors. This result is expected to inspire further investigation of BP and wider applications of BP in distributed estimation and control.