In this paper we explore the correspondence between four-dimensional Born-Infeld theory and five-dimensional classical gravity. The Born-Infeld theory side corresponds to the low-energy effective theory for open strings ending on coincident D3-branes in a (slowly varying) background Kalb-Ramond field, including all higher-derivative corrections. On the gravity side one has the gravitational (closed-string) description of D3-branes in the same background Kalb-Ramond field and the correspondence is thus a consequence of the open/closed-string duality. According to the correspondence the gravity side provides a description of the strong coupling limit of Born-Infeld theory. This is a correspondence between effective theories in a similar sense as in the fluid/gravity correspondence. We match the Born-Infeld and gravity sides up to, and including, two-derivative corrections. To this end, we find a new gravity solution for D3-branes with flat embedding in the background of an arbitrary constant background Kalb-Ramond field and show that there are no two-derivative corrections to this for a slowly varying Kalb-Ramond field.