Chaum [1,2] suggested a simple and efficient protocol aimed at providing anonymity in the presence of an adversary watching all communication links. Chaum's protocol is known to be insecure. We show that Chaum's protocol becomes secure when the attack model is relaxed and the adversary can control at most 99% of communication links. Our proof technique is markedly different than previous work. We establish a connection with information theory - a connection we believe is useful also elsewhere, and which we believe supplies the correct language to attack the problem. We introduce "obscurant networks" - networks that can obscure the destination of each particular player, and we show almost all executions of the protocol include such a network. The security guarantee we supply is very strong. It shows the adversary learns almost no information about any subset of players. Remarkably, we show that this guarantee holds even if the adversary has a-priori information about communication patters (e.g., people tend to speak less with those who do not understand their language). We believe this is an important issue in the real world and is a desirable property any anonymous system should have.