This Letter discusses the synchronization between two chaotic dynamical systems of different order, using an adaptive control scheme. The problem is closely related to the synchronization of strictly different chaotic systems. We show that the dynamical evolution of a fourth-order system can be synchronized with the canonical projections of a third-order system. In this sense, it may be said that the synchronization is achieved in reduced order, where by order we means the number of first order differential equations. The mathematical stability analysis is derived from the Lyapunov stability theory. Numerical simulations are presented to show the effectiveness and feasibility of the proposed scheme.