This paper studies global adaptive control of non-linearly parameterized systems with uncontrollable linearization. Using a new parameter separation technique and the tool of adding a power integrator, we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation. In contrast to the existing results in the literature, a key feature of our adaptive regulator is its minimum-order property, namely, no matter how big the number of unknown parameters is, the order of the dynamic compensator is identical to one, and is therefore minimal. As a consequence, global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one-dimensional adaptive controllers, without imposing any extra (e.g., convex/concave) conditions on the unknown parameters.