The paper considers the Kalman-Bucy filter for a linear system when the measurement noise covariance matrix is singular. It is shown that the problem of infimizing the square of a linear functional of the state estimation error is the dual of the optimal singular linear regulator problem. Furthermore there is an optimal reduced-order Kalman-Bucy filter for minimization of the trace of the state error covariance matrix, when all extremal controls for a dual regulator have finite order of singularity, and no Luenberger observer is needed. The proof is constructive. Necessary and sufficient conditions for the existence of a reduced-order optimal estimator are derived.