This paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle. Lagrange interpolation on a general two-dimensional nodal set is employed and, by imposing the boundary conditions weakly through a penalty term, the scheme is proven stable in L-2. This result is established for a general unstructured grid in the triangle. A special case, for which the nodes along the edges of the triangle are chosen as the Legendre Gauss-Lobatto quadrature points, is discussed in detail. The eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an O(n(-2)) restriction on the time-step when considering explicit time-stepping. (C) 1999 Elsevier Science S.A. All rights reserved.