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This paper considers the joint design of a multiple-input multiple-output (MIMO) radar with co-located antennas and a MIMO communication system. The degrees of freedom under the designer's control are the waveforms transmitted by the radar transmit array, the filter at the radar array and the code-book employed by the communication system to form its space-time code matrix. Two formulations of the spectrum sharing problem are proposed. First, the design problem is stated as the constrained maximization of the signal-to-interference-plus-noise ratio at the radar receiver, where interference is due to both clutter and the coexistence structure, and the constraints concern both the similarity with a standard radar waveform and the rate achievable by the communication system, on top of those on the transmit energy. The resulting problem is nonconvex, but a reduced-complexity iterative algorithm, based on iterative alternating maximization of three suitably designed subproblems, is proposed and analyzed. In addition, the constrained maximization of the communication rate is also investigated. The convergence of all the devised algorithms is guaranteed. Finally, a thorough performance assessment is presented, aimed at showing the merits of the proposed approach.