Sufficient conditions for existence and uniqueness of solutions for a coupled system of homogeneous equations defining dynamics of the gain and observer for ODEs obtained as a Galerkin projection of homogeneous PDEs are proposed. The conditions rely upon fundamental concept of uniform complete observability which is also used to design an exponentially convergent observer. Convergence of the observer is confirmed by numerical experiments with ODEs obtained from a hyperbolic PDE in 1D (Burgers-Hopf equation).