Explicit boundary dissipative conditions are given for the exponential stability in L^2-norm of one dimensional linear hyperbolic systems of balance laws ∂t ξ + Λ∂xξ −Mξ = 0 over a finite interval, when the matrix M is marginally diagonally stable. The result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised Saint–Venant–Exner equations.