This paper constructs the first example of a singular, abnormal minimizer for the Lagrange problem with linear velocity constraints and quadratic definite Lagrangian, or, equivalently, for an optimal control system of linear controls, with Ic controls, n states, and a running cost function that is quadratic positive-definite in the controls. In the example, k = 2, n = 3, and the system is completely controllable. The example is stable : if both the control law and cost are perturbed, the singular minimizer persists. Its importance is due, in pari, to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature. There, the problem is called the problem of finding minimizing sub-Riemannian geodesics, and it has been claimed that ail minimizers are normal Pontryagin extremals [The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962]. (If the number of states equals the number of controls, then the problem is that of finding Riemannian geodesics.) The main difficulty is proving minimality. To do this, the length (cost) of the abnormal is compared with all competing normal extremals. A detailed asymptotic analysis of the differential equations governing the normals shows that they are all longer