This 14-month study will enroll individuals with AAT deficiency. Participants currently using AAT protein replacement will discontinue its use for 19 weeks during the study. Participants will first attend a baseline study visit, which will include a medical history review; a physical examination; an electrocardiogram (ECG) to record heart activity; blood, urine, and semen collection; pulmonary function tests; and chest and arm scans. Participants will then attend a 5-day inpatient visit, during which they will receive a series of injections consisting of one of four different doses of rAAV1-CB-hAAT. Physical examinations will occur on all 5 inpatient days; pulmonary function testing, arm circumference measurements, and collection of blood, urine, and semen will occur on selected days of the inpatient stay. Follow-up study visits, with possible overnight stays, will occur on Days 14 and 90. On Days 30, 45, 60, 75, 180, 270, and 365, participants will have blood drawn at a local clinic. On these same days, study staff will contact participants by telephone to review their medical history and symptoms. Unused blood and semen samples will be frozen and stored for future research purposes. Participants will have yearly follow-up evaluations by either telephone or mail for a total of 5 years.