The treatment of anorexia nervosa often proves to be difficult. There are no drugs that work specifically for the treatment of anorexia nervosa. Experimental administration of metreleptin (synthetically produced leptin) to patients with anorexia nervosa has produced positive results. This study tests the effect of metreleptin in comparison with placebo, which could potentially make treatment easier. The aim of the study is to investigate whether treatment with metreleptin can help to reduce the symptoms of anorexia nervosa and improve mood and weight.