We propose a closed-loop spatial multiplexing system with limited feedback and rate control over MIMO fading channels. The transmitter adaptation is designed to maximize spectral efficiency subject to an average bit error rate (BER) constraint by adapting the number of substreams based on a given threshold, and a fixed constellation, which determines the instantaneous rate. The proposed system chooses the substream selection threshold as a function of the slowly changing average SNR. We analyze the performance of the proposed system by deriving closed-form expressions for spectral efficiency, average transmit power, and BER. Compared with the rate adaptive orthogonal STBC requiring highly complex constellation adaptation, the proposed rate control method achieves 5 dB power gain for a spectral efficiency of 7 bits/s/Hz at average BER of 10-4, with 2 transmit antennas.