We provide a new condition for testing the stability of a single-parameter, polynomially-dependent linear system of polynomial degree N of the form (x) over dot = A(rho) x, A(rho) Sigma(i=o) rho(i) A(i) (1) over a compact interval. The test is nonconservative and can be cast as a convex feasibility problem in terms of a pair of linear matrix inequalities (LMIs).