To construct numerical schemes of the Godunov type for solving magnetohydrodynamical (MHD) problems, an approximate method of solving the MHD Riemann problem is required in order to calculate the time-averaged fluxes at the interfaces of numerical zones. Such an MHD Riemann solver is presented here which treats all waves emanating from the initial discontinuity as themselves discontinuous. Thus shock jump conditions are used for rarefactions, which limits the applicability of this work to weak rarefactions, the case most important for computation. The solutions from our approximate MHD Riamnn solver consist of two fast waves (either shock or rarefaction) two rotational discontinuities, two rarefaction waves (either shock or rarefaction), and one contact discontinuity for a general MHD Riemann problem. In order to display rotational discontinuities, a three-component model is necessary. Only under very limited circumstances is there no rotational discontinuity involved and thus the two component approximation may be used in the MHD Riemann problem. The solutions of the MHD Riemann problem in the shock tube problem which generates the compound wave in the earlier work contain two fast rarefaction waves, two slow shocks, one contact discontinuity, and one rotational discontinuity in our formalism.