Does Treatment With GLP-1 Reduce Alcohol Intake in Patients With Alcohol Dependence?

Inclusion Criteria: * Informed oral and written consent * Diagnosed with alcohol dependence according to the criteria of International Classification of Diseases (ICD) 10, World Health Organization and DSM-5 * Alcohol use disorder identification test (AUDIT) score \>15 * Age 18 - 70 years * Heavy alcohol drinking defined as having alcohol consumption over 60 g of alcohol per day (men) or 48 g of alcohol per day (women) for at least 5 days in the past 30 days prior to inclusion measured by the TLFB method. Exclusion Criteria: * Severe psychiatric disease, defined as a diagnosis of schizophrenia, paranoid psychosis, bipolar dis-order 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 * Present or former diagnosis of type 1 or type 2 diabetes or plasma Haemoglobin A1c (HbA1c ) ≥48 moll/L at inclusion * Females of child bearing potential who are pregnant, breast-feeding or have intention of becoming pregnant within the next 9 months (26 weeks plus three months after discontinuation of Bydureon®) , 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 - IUD, IUS, bilateral tubal occlusion, vasectomised partner, sexual abstinence) (33) * Pregnancy (serum hCG \> 3 at inclusion) Impaired hepatic function (liver transaminases \>3 times up-per normal limit) * Impaired renal function (eGFR \< 50 ml/min and/or microalbuminuria) Impaired pancreatic function (any history of acute or chronic pancreatitis and/or amylase \> 2 times upper limit) * S-triglycerides \> 10 mmol/l * Former medullary thyroid carcinoma (MTC) and/or family history with MTC and/or Multiple Endo-crine 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 dependence including disulfiram, naltrexone, acamprosate and nalmefene or treatment with any of these compounds within 1 month prior to inclusion * Concomitant pharmacotherapy with dopamine active drugs, such as some types of Attention Deficit Hyperactivity Disorder (ADHD) medication (methylphenidate) * Receiving any investigational drug within the last 3 months * Use of weight-lowering pharmacotherapy within the preceding 3 month * Any other active substance use defined as a DUDIT-score \> 6 (for men) \>2 (for women) and fulfilling the criteria's for dependence of the substance according to the criteria of International Classification of Diseases (ICD) 10 (except nicotine) * BMI \<18.5 kg/m2 * Hypersensitivity to the active substance or to any of the excipients * Only for patients undergoing brain scans: Contraindications for undergoing an fMRI scan (magnetic implants, pacemaker, claustrophobia etc.). Contraindications for undergoing a SPECT-scan (allergy towards iodine, radiation exposure, excluding background radiation but including diagnostic x-rays and other medical exposures, exceeding 10 mSv in the last 12 months) * Unable to speak and/or understand Danish * Any condition that the investigator feels would interfere with trial participation