The minimum sum-of-squared error clustering problem is shown to be a concave continuous optimization problem whose every local minimum solution must be integer. We characterize its local minima. A procedure of moving from a fractional solution to a better integer solution is given. Then we adapt Tuy's convexity cut method to find a global optimum of the minimum sum-of-squared error clustering problem. We prove that this method converges in finite steps to a global minimum. Promising numerical examples are reported.