This trial is testing if prazosin, a medication typically used to treat high blood pressure, is effective in treating alcohol withdrawal symptoms and helping people with alcohol use disorder abstain from drinking. Patients will be randomly assigned to either take prazosin or a placebo pill for 12 weeks, and then will be monitored for 1- and 3-months after treatment. The primary outcome measures are the percentage of subjects who have no heavy drinking days, and the percentage of any drinking and heavy drinking days.