We consider a conservation law with strictly positive wave velocity and study the well-posedness of a suitable notion of solution for the associated initial value problem under a flux constraint active in the half-line R+. The strict positivity of the wave velocity allows for the dynamics in the unconstrained region R- to be fully determined by the restriction of the initial data to R-. On the other hand, the solution in the constrained region is dictated by the assumption that the total mass of the initial datum is conserved along the evolution. We formulate the transmission condition at the interface {x=0} in such a way that the boundary datum for the initial boundary value problem posed on R+ is given by the largest incoming flux that is admissible under the constraint, while the exceeding mass is accumulated in a “buffer” (as an atomic measure concentrated at the interface).