We consider the inertialess planar channel flow of a White-Metzner (WM) fluid having a power-law viscosity with exponent n. The case n = 1 corresponds to an upper-convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength k(-1). We find numerically that if n < n(c) approximate to 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to n(c), this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves became unstable and have the largest growth rate. If n exceeds n,, all disturbances are linearly stable. We consider asymptotically both the long-wave limit which is stable for all n, and the shortwave limit for which waves grow or decay at a finite rate independent of k for each n. The mechanism of this elastic shear-thinning instability is discussed.