We consider molecular communication, with information conveyed in the time of release of molecules. The main contribution of this paper is the development of a theoretical foundation for such a communication system. Specifically, we develop the additive inverse Gaussian (IG) noise channel model: a channel in which the information is corrupted by noise with an inverse Gaussian distribution. We show that such a channel model is appropriate for molecular communication in fluid media - when propagation between transmitter and receiver is governed by Brownian motion and when there is positive drift from transmitter to receiver. Taking advantage of the available literature on the IG distribution, upper and lower bounds on channel capacity are developed, and a maximum likelihood receiver is derived. Theory and simulation results are presented which show that such a channel does not have a single quality measure analogous to signal-to-noise ratio in the AWGN channel. It is also shown that the use of multiple molecules leads to reduced error rate in a manner akin to diversity order in wireless communications. Finally, we discuss some open problems in molecular communications that arise from the IG system model.