The author derives a differential Harnack inequality for weakly convex solutions of the mean curvature flow. This flow is the evolution of a hypersurface \(M^n\) in Euclidean space \(\mathbb{R}^{n + 1}\) by its mean curvature. As an application he shows that any strictly convex solution of the flow which is defined for all times and such that the mean curvature assumes its maximum value must be a translating soliton, i.e. a surface evolving by translating in space with constant velocity. For the proof the author derives from the mean curvature flow equation explicit equations for the evolution of the second fundamental form and the basic Harnack expression.