We investigate probabilistic propositional logic as a way of expressing and reasoning about uncertainty. In contrast to Bayesian networks, a logical approach can easily cope with incomplete information like probabilities that are missing or only known to lie in some interval. However, probabilistic propositional logic as described e.g. by Halpern [1], has no way of expressing conditional independence, which is important for compact specification in many cases. We define a logic with conditional independence formulae. We give an axiomatization which we show to be complete for the kind of inferences allowed by Bayesian networks, while still being suitable for reasoning under incomplete information.