A simple reduced-order adaptive filter, optimal in the sense of minimum prediction error, is proposed for estimating the state of high-dimensional systems in which the process and observation noise statistics are unknown. It is shown that implementation of this adaptive filter requires the solution of only two linear difference equations, the dimensions of which are the dimensions of the full and reduced states, respectively, and that no solution of either an algebraic Riccati equation or a Lyapunov equation is needed. In addition, substantial gain in computer memory and CPU time is obtained by parametrization of the filter gain in the form of the product of two matrices, one of which is a prescribed projection from the reduced space onto the full space. A twin experiment on data assimilation with a quasi-geostrophic ocean model shows the efficiency of the proposed approach.