Changes in a cell's external or internal conditions are usually reflected in the concentrations of the relevant transcription factors. These proteins in turn modulate the expression levels of the genes under their control and sometimes need to perform nontrivial computations that integrate several inputs and affect multiple genes. At the same time, the activities of the regulated genes would fluctuate even if the inputs were held fixed, as a consequence of the intrinsic noise in the system, and such noise must fundamentally limit the reliability of any genetic computation. Here we use information theory to formalize the notion of information transmission in simple genetic regulatory elements in the presence of physically realistic noise sources. The dependence of this "channel capacity" on noise parameters, cooperativity and cost of making signaling molecules is explored systematically. We find that, in the range of parameters probed by recent in vivo measurements, capacities higher than one bit should be achievable. It is of course generally accepted that gene regulatory elements must, in order to function properly, have a capacity of at least one bit. The central point of our analysis is the demonstration that simple physical models of noisy gene transcription, with realistic parameters, can indeed achieve this capacity: it was not self-evident that this should be so. We also demonstrate that capacities significantly greater than one bit are possible, so that transcriptional regulation need not be limited to simple "on-off" components. The question whether real systems actually exploit this richer possibility is beyond the scope of this investigation.