Dengue is an infection of increasing global importance, yet uncertainty remains regarding critical aspects of its virology, immunology and epidemiology. One unanswered question is how infection is controlled and cleared during a dengue infection. Antibody is thought to play a role, but little past work has examined the kinetics of both virus and antibody during natural infections. We present data on multiple virus and antibody titres measurements recorded sequentially during infection from 53 Vietnamese dengue patients. We fit mechanistic mathematical models of the dynamics of viral replication and the host immune response to these data. These models fit the data well. The model with antibody removing virus fits the data best, but with a role suggested for ADCC or other infected cell clearance mechanisms. Our analysis therefore shows that the observed viral and antibody kinetics are consistent with antibody playing a key role in controlling viral replication. This work gives quantitative insight into the relationship between antibody levels and the efficiency of viral clearance. It will inform the future development of mechanistic models of how vaccines and antivirals might modify the course of natural dengue infection.