This article deals with the output regulation of a nonlinear Korteweg–de Vries (KdV) equation subject to a distributed disturbance. The control input and the regulated output are located at the boundary. To achieve this objective, we follow a Lyapunov approach. To this end, inspired by a strictification methodology recently introduced in the finite-dimensional context, we construct an ISS Lyapunov functional for the KdV equation thanks to the use of an observer which is designed following the backstepping approach. Then, thanks to this Lyapunov functional, we apply the forwarding approach in order to solve the desired output regulation problem.