The short answer is most likely not, but it’s possible depending on whether it is a felony or misdemeanor drunk driving conviction. For felony DUI, it is a definite no – in fact the mere possession of a firearm is prohibited. For misdemeanor DUI, as long as a certain time period has passed since the conviction (in most cases eight years), then there is the possibility that an individual can be eligible for a Concealed Weapons License.