Complete disturbance rejection problems are equivalent to zeroing (cancelling) all the Markov parameters of the closed loop system between the disturbance and the controlled output. When this is not possible, one might consider partial disturbance rejection which can be defined as zeroing the first, say k, Markov parameters. In this article, our objective is to present general solvability conditions for the partial disturbance rejection problem by dynamic output feedback under the constraint of internal stability. With this solution we also obtain a suitable parametrization for the set of all solutions of the problem which is then used to obtain an H-infinity norm bound on the closed loop system. In the first part of the paper, a natural framework for the partial disturbance rejection problem is introduced. This framework consists of the ring of stable and proper rational functions and its quotient rings. Thus, the solvability conditions and the set of all solutions to the problem are easily obtained. The parametrization of the set of all solutions provides an opportunity to pursue further design goals. Along this line, H-infinity minimization has been incorporated into the problem. The upper and lower bounds on the H-infinity norm of the closed loop transfer function is obtained and compared with direct H-infinity disturbance attenuation. The results are illustrated with a simple example.