Obesity is associated with an increased risk of severe Covid-19. However, the effectiveness of SARS-CoV-2 vaccines in people with obesity is unknown. Here we studied the relationship between body mass index (BMI), hospitalization and mortality due to Covid-19 amongst 3.5 million people in Scotland. Vaccinated people with severe obesity (BMI>40 kg/m2) were significantly more likely to experience hospitalization or death from Covid-19. Excess risk increased with time since vaccination. To investigate the underlying mechanisms, we conducted a prospective longitudinal study of the immune response in a clinical cohort of vaccinated people with severe obesity. Compared with normal weight controls, six months after their second vaccine dose, significantly more people with severe obesity had unquantifiable titres of neutralizing antibody against authentic SARS-CoV-2 virus, reduced frequencies of antigen-experienced SARS-CoV-2 Spike-binding B cells, and a dissociation between anti-Spike antibody levels and neutralizing capacity. Neutralizing capacity was restored by a third dose of vaccine, but again declined more rapidly in people with severe obesity. We demonstrate that waning of SARS-CoV-2 vaccine-induced humoral immunity is accelerated in people with severe obesity and associated with increased hospitalization and mortality from breakthrough infections. Given the prevalence of obesity, our findings have significant implications for global public health.