We propose a novel transceiver design technique to facilitate flexible spectrum sharing between a multiple-input multiple-output (MIMO) radar and a full-duplex (FD) MIMO cellular system. The optimization problem for maximizing the rate of the cellular system is formulated, subject to the constraints of individual power at the uplink users, total power at the base station, and interference power towards the MIMO radar from the cellular system so that the detection probability of the radar is not hindered. We show that the above problem can be cast as a second-order cone programming problem and the joint design of transceiver matrices can be obtained through an iterative algorithm. Numerical results show that using the spectrum shared by the radar, the FD cellular system can achieve sum rate of up to 25-30 bits/sec/Hz for a reasonable self-interference cancellation of around -70 dB. However, to facilitate this, while also maintaining a detection probability of around 0.9, the radar needs to spend an extra power of around 2-3 dB.