Sufficient conditions for fixed-time convergence of matrix differential Riccati equations towards an ellipsoid in the space of symmetric non-negative matrices are proposed. These conditions are based on the classical concept of uniform complete observability. The fixed-time convergence is demonstrated for the Riccati matrix and its inverse. This convergence is then used to design a globally convergent observer for bilinear chaotic differential equations (e.g. equations with zero Lyapunov exponents). Convergence of the observer is confirmed by numerical experiments with ODEs obtained by finite-difference discretization of a hyperbolic PDE in 1D (Burgers-Hopf equation).