Does Semaglutide Reduce Alcohol Intake in Patients With Alcohol Use Disorder and Comorbid Obesity?

Inclusion Criteria: * Informed oral and written consent * Diagnosed with alcohol dependence according to the criteria of the International Classification of Diseases 10 (ICD-10), and diagnosed with alcohol use disorder according to the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5) * Alcohol use disorder identification test (AUDIT) score \>15 * Body mass index (BMI) above or equal to 30 kg/m2 * Age 18 - 70 years (both included) * Heavy alcohol drinking defined as more than 6 days with alcohol consumption over 4 units (48 g alcohol) for women and 5 units (60 g alcohol) for men during a consecutive 30-day period, within 40 days prior to baseline evaluation, measured by the TLFB method. The 30-day period will be the 30 consecutive days with the biggest alcohol intake (most heavy drinking days and the largest amount of total alcohol) out of the 40 days. Exclusion Criteria: * Severe psychiatric disease, defined as a diagnosis of schizophrenia, paranoid psychosis, bipolar disorder or mental retardation * A history of delirium tremens or alcohol withdrawal seizures * No serious withdrawal symptoms at inclusion (a score higher than 9 on the Clinical Institute Withdrawal Assessment of Alcohol Scale, Revised (CIWA-Ar)) at baseline examinations * Present or former neurological disease, including traumatic brain injury * Type 1 diabetes, type 2 diabetes in poor glycaemic control (defined as HbA1c ≥48 mmol/l or fasting plasma glucose above 7.0 mmol/l at inclusion) * Females of childbearing potential who are pregnant, breast-feeding or have the intention of becoming pregnant within the next 9 months (26 weeks plus two months after discontinuation of semaglutide), or are not using contraceptives (during the whole study period) considered as highly effective (combined (oestrogen and progestogen containing) hormonal contraception associated with inhibition of ovulation (oral, intravaginal, transdermal), progestogen-only hormonal contraception associated with inhibition of ovulation (oral, injectable, implantable) intrauterine device, bilateral tubal occlusion, vasectomised partner, sexual abstinence). * Pregnancy (serum human chorionic gonadotropin (hCG) \> 3 U/L at inclusion) * Impaired hepatic function (liver transaminases \>3 times the upper limit) * Impaired renal function (eGFR \< 50 ml/min and/or plasma creatinine \>150 μmol/l) * Impaired pancreatic function (any history of acute or chronic pancreatitis and/or amylase \> 2 times upper limit) * Former medullary thyroid carcinoma (MTC) and/or family history with MTC and/or Multiple Endocrine Neoplasia syndrome type 2 (MEN 2) * Cardiac problems defined as decompensated heart failure (NYHA class III or IV), unstable angina pectoris and/or myocardial infarction within the last 12 months * Uncontrolled hypertension (systolic blood pressure \>180 mmHg, diastolic blood pressure \>110 mmHg) * Concomitant pharmacotherapy against alcohol use disorder, i.e., disulfiram, naltrexone, acamprosate, or nalmefene, since the first of the 30 drinking days registered for inclusion at the TLFB-schedule. * Receiving any investigational drug within the last three months * Use of weight-lowering pharmacotherapy within the preceding 3 months * Any other active substance use defined as a DUDIT-score \>1 (except nicotine) * Hypersensitivity to the active substance or any of the excipients * Only for patients undergoing brain scans: o Contraindications for undergoing an MRI scan (magnetic implants, pacemaker, claustrophobia, etc.) * Unable to speak and/or understand Danish * Any condition that the investigator feels would interfere with trial participation