In this paper we study the cognitive interference channel with a common message, a variation of the classical cognitive interference channel in which the cognitive message is decoded at both receivers. We derive the capacity for the semi-deterministic model, a class of channels in which the output at the cognitive decoder is a deterministic function of the channel inputs. We also show capacity to within a constant gap and a constant factor for the Gaussian channel. Most of these results are shown using an interesting transmission scheme in which the cognitive message, decoded at both receivers, is also pre-coded against the interference experienced at the cognitive receiver. The pre-coding of the cognitive message does not allow the primary decoder to reconstruct the interfering signal; the cognitive message acts instead as a side information at the primary receiver when decoding its intended message.