In this paper we present rigorous a posteriori L2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution-dependent (Online) calculation of the exponential-in-time stability factor by the Successive Constraint Method. Numerical results indicate that the a posteriori error bounds are practicable for reasonably large times — many convective scales — and reasonably large Reynolds numbers — O(100) or larger. Work in progress (in collaboration with Dr Paul Fischer) considered the full unsteady incompressible Navier-Stokes equation.