Explicit boundary dissipative conditions are given for the exponential stability in L-2-norm of one-dimensional linear hyperbolic systems of balance laws partial derivative(t)xi + Lambda partial derivative(x)xi - M xi = 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. (C) 2011 Elsevier Ltd. All rights reserved.