This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at $x = 0$. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely $x < 0$ or $x > 0$. At interface $x = 0$ we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove the existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non-uniqueness of both continuous and discontinuous (at interface $x = 0$) solutions. The behavior of some numerical scheme is also investigated.