Motion planning and control of a boundary controlled quasilinear parabolic partial differential equation in one spatial variable is considered. The approach relies on a flatness property of the system; namely, that the system solution can be differentially parameterized in terms of a flat output which, in the case considered, is a boundary value. Such a parameterization allows straightforward motion planning and computation of a control law. The approach is based on power series in the spatial variable, and the convergence of these series is ensured by choosing the flat output to be a nonanalytic, smooth function of appropriate Gevrey class.